Sidebar: I really am trying to get a handle on this whole gender thing, and understand what they are trying to say. I mean, it seems like a load of double talk and kidding themselves, but there's an awful lot of them, so I'm allowing that it's just me not getting it, and trying to see the POV. One or two superficial questions and they get all sagey and say it's complicated. That's pretty much as far as we get.
Good luck with your endeavor. I don't think you're ever going to be able to get a reasonable explanation in this context, not one that actually makes sense.
From my perspective, I take it from the feminist perspective. That perspective separates the social confines and expectations that are placed around a person's sex in a limiting and regressive way from the fact of their sex itself. Thus, a female person is female, no matter what. But all of the notions wrapped around that, of females being docile, subservient, quiet, of females being "naturally" suited to homemaking, to care giving, of females being unsuited for decision making and leadership, of females being dominated by their emotions and unable to employ logic or reason, the conflicting narrative of females being simultaneously required to be pure vessels in a virginal state but also being dried up prudes if they are, the entirety of the madonna-whore concept, the pervasive view of females as being "lesser males" whose primary value is based on how well they please males in terms of their availablity for titillation (even to the extent of females being expected to smile on demand because it makes males happy to see females smiling), their willingess to subdue their own needs in order to see to the desires of males and provide for male comfort, etc.
All of those things are what feminists coined "gender roles" and "gender expectations". Don't get me wrong, there are just as many wrappings placed around males, and some of them are even stricter than for females (for example, what clothing a male is allowed to wear and still be considered "male enough").
The feminist take is that all of those socially-constructed barriers and constraints are what comprises "gender". Our physical state, our bodies and the experiences of those bodies, those are "sex". Sex is a matter of reality, of fact; gender is a result of society - specifically a result of a male-dominated society that was constructed by males for the benefit of males.
That is my understanding of gender. So far as I can tell, that's also the concept of gender that underlies "gender identity". Gender identity implies an alignment with those restrictive social barriers, a desire to conform to those social shackles and to revel in them. When you start asking transgender people what "woman" means to them, it almost always comes back around to those stereotypes, those artificial boundaries placed on the sexes. And those stereotypes get held up as an ideal that defines the female sex, as if a male who feels alignment with those stereotypes is somehow a "kind of female" rather than a gender nonconforming male. (Same for the opposite direction).
That is my understanding of gender, and gender identity. I have yet to see anything from the transgender activist community that provides a meaningfully different perspective, one that is not a matter of alignment with stereotypes that are, by and large, harmful to females.
And by venerating those gender roles and gender expectations, by holding up the stereotypes as "what it means to be a woman" and the basis for how a male can be considered a "woman", the effect is to reinforce those stereotypes, and to embed them even deeper into our social fabric.
I know a few transsexual people (I chose that word intentionally, and it is how they refer to themselves) who do NOT reflect that view. These are individuals with severe and very real dysphoria about their sexed bodies. One of them explained it as "Every time I undress and look down, I'm startled to see a penis. It doesn't belong there, and I feel like it's not a part of me, it's an alien appendage".