Previously I've noted what I see as four categories:
* Physical sex, i.e. male, female, intersex. Whether the physical traits we're looking for are brain, genitals, hormones, chromosomes, etc., and whether there might be physical markers we aren't yet aware of or can't scientifically identify, is an additional layer of complexity.
* Gender feelings -- these are private unless/until you share them somehow
* Gender presentation -- how you're communicating your gender to others, e.g. what you wear
* Gender role -- whether your gender fits an existing social role, usually man or woman, or another option depending on what is culturally recognized
Your question -- biology or social construct? -- is important, but the answer doesn't have to be all-or-nothing. Some of our traits, subjective feelings, and social experiences have biological explanations. Others might be better explained in terms of culture and individual psychology. Often, both are relevant. If you want to describe, for example, a first-date interaction, you can describe it with reference to evolutionary biology (our physical sexes, brain and otherwise, are hard-wired, of course, and they are relevant when we go on dates) and you can describe the date in terms of cultural expectations about gender. Multiple things are happening in that interaction.
Gender is complex. If someone can get precise about their term, that is great. They can say "the sex I believe my brain to be" or "the social construct that I perceive exists." The difficulty here is that neither of these are tangible things. Does the human brain have a sex? Maybe not. (I'm aware that there are scientific studies about how men's brains and women's brains might behave differently, on average. But can I reliably identify a male or female brain, or even make an educated guess? No. I've never seen any human brain at all--dissected in a lab, I mean. And the only brain of which I have subjective, direct experience is my own.) And what does a social construct look like? How do we know when we've encountered a personal opinion and when it morphs into a collective belief? How do we know the social construct isn't rooted in biology, after all?
I think we haven't developed more specific terms because we're unable to use them reliably and precisely. I don't want to say "I'm a man because I have a male brain" because I do not, in fact, know that I have a male brain or even that there is such a thing as a male brain. I don't want to say "I'm a man because I fit an existing social construct" because I have no idea what construct I'd be talking about. Either way, these would be post-hoc rationalizations. The idea that I was "not a woman" and "maybe more like a man -- 'man' is a fine label" surfaced in adolescence, almost 30 years ago. Basically it has not changed. It feels inherent to the way I feel, think, and live. But I never knew, and I still don't know, whether it's about biology, construct, or what. So often I just say "gender" without trying to explain what gender is or what kind of gender I'm referring to. All the manifestations of gender--physical, feelings, presentation, role--are mixed together.
