"Woman" is a well defined term and has been for millions of years of combined human history and experience, across all cultures and civilisations.

Funny that you mention that, because there are several historical examples of transgender individuals throughout time, and across several civilisations.

Not feeling like you belong to a set called "man" doesn't mean you belong to the set called "woman"

That is true, but it's not like I just "flipped a switch" overnight and went "Oh, if not man, then woman." I ruminated on the subject for the better part of two months figuring out what the hell I was, experimenting with different masculine, feminine, and non-binary forms of expression until I figured out that being a woman was most comfortable for me.

Thankfully I've come to that conclusion, though it was difficult to accept it at first, considering the society I'm inserted in is transphobic by default, not helped by people preaching some twisted form of gender prescriptivism rooted in their own, static perceptions of masculinity and femininity. Because of that, I and several other people have to justify their right to exist unmolested in this world when we just want to express ourselves genuinely and be happy and comfortable in our own skins.

For real, mate, I'm not in the mood to argue that I am allowed to exist and be content with my own body.