While I will readily admit that I don't have the experience that the full writers do, I would like to offer some commentary that hasn't been offered yet. Unlike the majority of the writers who responded to your previous questions, I do understand the male perspective, and I think a response from that perspective ought to be provided.
I don't have as much context as I'd like to have--I obviously wasn't a writer at the time the original question was posted, and I was still being spoonfed Board functionality when its follow-up Board Question #92521 came into the inbox. With that said, an age-old proverb says that fools rush in where angels fear to tread, so having thus furnished my masculine credentials, I'll dive in and see if we can work something out of your question. I'm going to approach this topic presuming that you're asking in good faith, and I hope you'll reciprocate with my response.
The question: What specifically is said or done which demonstrates objectification?
Let's go all the way back to the beginning and take a look at Board Question #91428 for our model. Since this seems to be the comment which prompted the most ire and led to this chain of related questions, I'd like to go through it a piece at a time and examine the attitudes at play.
"So I’ve noticed the recent question about changing modesty standards. I was actually going to ask if it seems like BYU is getting more lax on what people where at certain athletic events. The female track athletes don’t seem to be as modest this year as years before...I noticed after the Rex lee run there were some women around afterwards that had nothing but a sports bra on top."
Right here at the outset, I have a concern. What's considered modest track wear? In what way is the track wear from the current year less modest? I ask this not to sound flippant, but because we need to know what it is that defines immodesty in the first place. The questioner (hereafter designated by his 'nym "Not Tom") doesn't say, which means it's difficult to get a helpful idea of what his idea of modesty is.
We don't really conceptualize modesty this way in the church often, but I would suggest that modesty is very much a contextual thing. What may be modest in one situation can be totally inappropriate in another. I would wholeheartedly agree that wearing only a sports bra would be wildly inappropriate at a work meeting, a stake fireside, or a formal dinner, but most everyone I know who runs dresses lightly because it's what's most comfortable and practical. During the blistering hot summers on my mission in the southern United States, missionaries often joked that it must be a legal requirement for state residents to remove their shirts before going outside. There's nothing wrong with adjusting your dress to fit the climate. So long as nobody is violating laws surrounding public indecency and exposure, I see no reason why a woman wearing a sports bra shouldn't be considered identically to a man running topless: someone who is dressing comfortably for exercise in the heat should be avoiding as many extra layers as is practical. So, while Not Tom makes no explicit assertions about modesty, the underlying assumption here seems to be that anything less than a shirt, or another top covering the bra, is inherently immodest. That isn't something I agree with.
"...It just seems that we’re becoming less and less destinct [sic]. Like do you think it’s immodest for a woman to run without a shirt? Like I think it’s crazy when in years past you might hear someone say the cheerleaders are immodest. But Don’t you think it sends a certain message when a woman doesn’t have a shirt on?"
As I said above, modesty is contextual. So no, I don't believe it's inherently immodest for a woman to run without a shirt on. I do think it would be immodest for a woman to run totally topless, but that's not the question being asked. I don't think running in a sports bra sends a particular message other than the woman in question has enough sense to dress comfortably for the heat. Again Not Tom's assumption here seems to be that women are dressing immodestly if they're not wearing a top of some kind, with no apparent regard to other factors playing into their choice of dress. The equivocation in that assumption is clear in the two different questions being asked as though they're actually one and the same: first we're asked whether it's immodest for a woman to run without a shirt. Then, in support of that question, we're asked, are women not sending a certain message by being shirtless? Indeed, how could one suggest that this sort of thing isn't immodest?
The best answer is something more like "Well, the message being sent really depends on the situation, so your second question doesn't actually follow from your first," but this option doesn't even seem to be in view in the original question, which is a little concerning. Instead, modesty is a binary decision, simply equivalent with shirtlessness. This may be an easy framework of modesty, but it's not a sufficient one, for reasons we're about to get into. But before we move on to the end of the question, let's get a working definition of modesty so we're on the same page.
Latter-day Saint philosopher Blake Ostler has an absolutely incredible article on the nature of relationships with God and others in which he discusses, among other things, what it means to be in a proper relationship with others as people (as "Thous," to use his terminology) rather than mere objects (or "Its"). The entire article is well worth reading, but I'll reproduce the section which seems most relevant to our discussion of objectification and modesty:
"To speak to a Thou in a proper relationship is not to use one's vocal cords but to stand before existence and to relate to it in a certain way; to take an ethical stance in relation to persons. To treat a person as an object is to treat it as an It, to regard the object, even if it be a He or a She, as if it were a mere thing. We stand apart from an object in order to coldly scrutinize and exploit it: to observe, measure, categorize, and manipulate it - to bend it to our advantage. In an I-It relationship there is no genuine reciprocity. The relation is that of manipulator to an instrument, of mechanic to engine, or computer scientist to computer. To treat a person as an It implicitly results in our being treated as mere objects. Thus, for me to speak to a person as a mere object also objectifies me as an object, a thing rather than a person in relationship. If I treat a person as a mere object of pleasure, to be appropriated for my purposes and to deny any independent purpose to that person, then both of us have lost our intrinsic value as Thou in a relationship. The relation is one of a person observing a pornographic picture of a person. Both the observer and the observed are mere objects in such a relation. It is the relation of a biological organism controlled by its hormones to a mere body."
Good stuff. Keep it in mind; we'll come back to it as we treat the last part of the question.
"If you’re a woman reading this I don’t think you understand what it’s like to be a guy seeing a not fully dressed woman. It’s getting tougher and tougher being a guy in our society weather [sic] it’s being on social media, or at the beach, or just out and about in public."
This is where the original question goes from misguided in its unconscious assumptions to rather patronizing. This is also where those assumptions are most clear.
It is true that women don't and can't have precisely the same experience that men do. At risk of stating the obvious, though, experiencing feelings of attraction and romantic or sexual desire is not exclusive to men. Not Tom's assertion may be well-intentioned, but it's both reductive and flatly wrong. As guppy of doom astutely pointed out in her original response to this statement, "...I don't know what it's like for men to see a woman in a sports bra. But I know how I feel when I see tan, fit men running in nothing but shorts down the road. I've gotten really good at...appreciating his physique, recognizing the attraction, and moving on. I intentionally look away and don't let my thoughts wander...Society (especially Mormon society) seems to have a view that women don't have a sex drive. But we do. It feels limiting and almost dehumanizing to say that women don't have this fundamental human aspect that is crucial for the continuation of our species."
I'm not a woman, but I'm willing to bet the sexual experience is not so distinct between the sexes that our shared moments of "appreciating [the] physique, recognizing the attraction, and moving on" are foreign concepts. I know what experiencing sexual attraction feels like, even though I've never had the slightest attraction towards the tan, fit men I see at the apartment pool every weekend. I'm not the least bit interested in other men, but having been attracted to women, I can at least reasonably try to approximate what a girl is feeling when she describes a man as attractive.
Further, Not Tom goes on to link this experience of seeing "a not fully dressed woman" with the difficulty of being male in today's society, which continues the same underlying assumption that women are exempt from the difficulty of sexual temptation and that this is primarily or even exclusively a male problem. I know this is a common way to look at it, and I know discourse on modesty in the church has conditioned many of us to do exactly this, but it's still a wrongheaded and harmful assumption. At best, it perpetuates inaccurate stereotypes of women (especially Latter-day Saint women) as demure, sexless creatures, who presumably feel nothing at the sight of not fully dressed men. At worst, it actively denies the notion that women have any kind of sexual needs or sexual agency at all, and isolates and shames those women who do struggle with sexual sin. I point this out not to accuse Not Tom of deliberately doing all or even any of those things, but in response to the current question of "what is said or done which demonstrates objectification?" I would point to statements like these as examples. Suggesting that sexual impropriety is a male problem, something too arcane or masculine to be understood by women, belittles the breadth of the female experience. Back to guppy of doom again, who by virtue of her femaleness probably makes this point better than I: "It feels limiting and almost dehumanizing to say that women don't have this fundamental human aspect that is crucial for the continuation of our species."
I know some female friends who have been in Young Women's/Relief Society lessons where the intended lesson on pornography or sexual purity was simply ignored, because it's so taken for granted that "good girls" don't even have the desire to sexually misbehave that it's not even taken seriously. How would you feel to be told by a friend or authority figure that your struggle is one that your sex as a whole should never even be experiencing? Awful is no word for it. It's inexcusable. Not Tom hasn't said or done those specific things in the original question and I'm not suggesting that he has, but the same fundamental attitude is reflected in the kinds of questions that he is asking.
Now, with all this said, let's go back to the main point of the statement. It's hard to be a man in society today and be surrounded by women who don't always live up to Latter-day Saint standards of modesty. I'm a man. I'm sympathetic to the idea. I get it. But this line of thinking is still fatally flawed. Let me tell you a story.
Years ago, I was reading through posts from Fight the New Drug on Facebook, and they shared an image of a couple proudly wearing their Porn Kills Love t-shirts, as they often do. I noticed in the comments one man who was quite apart from the usual flood of positivity in the comments. He was instead grumbling that even Fight the New Drug, of all places, didn't have better sense than to post suggestive if not actually pornographic advertising, and he was prepared to stop supporting the organization over their blatant and tragic hypocrisy. After much confused pressing from fellow readers, it became clear that he was disgruntled because the woman in the photo was wearing form-fitting yoga pants, and he blamed Fight the New Drug for failing to recognize that posting such photos was essentially providing their own lighter and softer form of pornography as far as he was concerned.
Does that seem a little absurd to you? I certainly thought this man's complaint was a little ridiculous. Unfortunate for that man, and sad, yes--but quite frankly, a little absurd. In my view, there was nothing remotely objectionable about the photo at all. Yet here this man was, insisting that a Facebook page dedicated to opposing pornography was actually aiding and abetting his own addiction to the same because of the thoughts that entered his mind.
The problem with this man's thinking is that he has given up his own agency in the situation. Let us turn to one of the greatest works of the Disney Renaissance, The Hunchback of Notre Dame, for insight. Specifically, let's look at some of the lyrics to "Hellfire," which, quite aside from its illustrious status as one of the best songs ever composed for an animated Disney film, is directly relevant to our question:
It's not my fault
I'm not to blame
It is the gypsy girl
The witch who sent this flame
It's not my fault
It's in God's plan
He made the Devil so much stronger than a man
Frollo is, while tragically mistaken, not entirely wrong. It is true, after all, that he cannot entirely control which thoughts and feelings enter into his mind as a result of Esmeralda's infamous performances as a dancer. None of us is in a position to precisely control our every thought, feeling, and inclination. However, where he ultimately falls is in his defeatist insistence that he must therefore be utterly powerless against his feelings--even going so far as to conclude that God Himself must in some way be responsible for them and thus it is useless for him to reject them. This is fundamentally in conflict with the teachings of the restored gospel, which enthrone agency as the single most important gift of God to man. Short of psychological or physiological damage to the point of addiction or disability, we are never deprived of our agency, no matter what stimuli we're presented with.
I hear you now. "Stop rambling, 9S! What in blazes does this have to do with objectification?" Remember what Ostler said a few paragraphs up? To give up one's agency--to shift responsibility to the attractive woman wearing a low-cut top, or a sports bra, or a skimpy bikini--is to reduce oneself to an object to be acted upon, and in so doing, to also objectify the woman on the other end of the relation. Both of you are denied intrinsic value in that relation. Ask yourself this: are you an intrinsically valuable son (or daughter) of God, striving to have virtuous thoughts in spite of the unfortunately objectionable clothing or behavior you might see from other intrinsically valuable sons or daughters of God? Or are you a mere biological organism, enslaved by your hormonal responses to the bodies or the clothing of other biological organisms? Setting aside the question of sports bras, even when others are dressed entirely modestly, we will still be presented with thoughts that tempt us. The same yoga pants I didn't even notice were comparable to pornography for another man. A pretty sister missionary whom my mission companions thought was unremarkable was enough to occupy my thoughts for hours if I wasn't careful. These are all instances of objectification. Some are more mild than others, and many times the cause is a natural part of our being human, but in every case the responsibility to recognize the behavior and turn away from it begins with us, not the other person. That is precisely what the scriptures mean when they admonish us to put off the natural man.
In conclusion, and to come all the way back to your question, Tom, I think a few things can profitably said about objectification:
It doesn't always happen consciously. Often, we're merely acting out the patterns that we've been conditioned to, as when we suggest that women are responsible for men's thoughts. Most people accept this statement uncritically and with the best of intentions, without realizing that it's actually harmful to both men and women. It suggests that men are helpless objects to be acted upon, and that women are valued by the influence they exert on men rather than their own intrinsic value as daughters of God.
It doesn't mean we're bad people. All of us have sinned and fall short of the glory of God, and all of us encounter the temptation to objectify another person, whether we're like Frollo refusing to take responsibility for our lustful thoughts or we're simply manipulating people to gain a financial, social, or emotional advantage. The key is recognizing where we've gone wrong and working to remedy it.
It doesn't have to be exclusively sexual in nature. Although it's most commonly discussed in connection with sexuality, objectification is really about agency and personhood, not about clothing or feelings of attraction. When we deny that we have the power to respond to our situations, or limit the experiences or understanding of others in their situations, we're falling into the trap of defining them by what they are (or are not) and looking past their intrinsic value as a Thou rather than an It.
I hope something in this is genuinely helpful to you. I wish you the best.