Good question.
I would be one of those that want to look female and be treated as female in public. I dress in female clothing, and am on hormone therapy (just spiro and finasteride at this point). But I don't identify as woman. I think if you look at gender identity, presentation, behaviour and roles as potentially seperate things it makes it much easier to understand.
I've tried seeing myself as a woman inside but it just does not seem to fit, I have a non binary gender identity as far as I can tell. Weird eh? I don't currently have any intention of getting rid of my man thing and don't find it a bother, except in hotpants. Not that I really consider myself androgyne anymore, that does not quite fit either but I think I fit in with this crowd being non-binary transgender. I'm open to the fact that this could change. You never know I could be a woman hiding from myself, but I doubt it. My gender identity seems pretty solid.