Although I appreciate the complexity / fuzzines of the problem, and the additional challenges of having these conversations amid an often unfriendly or even dangerous political / social landscape, I still think it's helpful to provide a slightly better answer than your conclusion here.

Fundamentally, of course, the real problem may be that we are trying to categorize ourselves or others at all.

But if we do decide to apply a lable (i.e. 'woman' or 'man') to ourselves, I think there is a problem with refusing to give it some actual shape.

Words exist to communicate - which means at least two people need to understand what a word means. One lone person doesn't need a word for themself if they're not planning to describe themself to someone else.

If a person thinks the word 'woman' describes herself better than 'man', at some level, it's because they've heard other people use both words, and they believe that the way people use the word 'woman' fundamentally describes them better.

A cis woman, for example, might simply say that she has a uterus, an she knows she is a woman for that reason. Whereas a trans woman might say that she always felt some disconnect with her body, or with gender roles placed on her by a society that didn't call her a woman (or any number of other reasons).

Zhenya has, I think, a really good attempt at a definition here: https://medium.com/@cde.zhenya/simple-non-circular-non-transphobic-definitions-of-gender-c1e288d49cd5

I think it's still a good idea to keep in perspective that a person is who they are regardless of the word. But if you are going to use a word, it's good to have some reason for using it (which is ultimately what a definition provides) in your back pocket.

Does that make sense?