I've only ever seen two non-biologically based accounts of what a woman is.
One is "A woman is someone who asserts they are a woman" which seems, a bit, y'know, circular.
The other is "A woman is someone who adheres to society's view of what constitutes appropriate feminine appearance, dress, behaviour and mannerisms." Which is both insulting and fucking useless, because by this reckoning (as an adult human who likes playing football) I'd be a woman in the USA and a man over here.
Seriously - you have a definition which works fine 99.9% of the time, and for the other 0.1% there are ways of looking into the person's underlying biology more carefully, in order to establish what sort of DSD (difference of sexual development) applies in their case. It has explanatory power - it tells us how human reproduction occurs, how it intersects with evolutionary change in our species, what's similar between us and other mammals, and different between us and true hermaphrodites (eg. earthworms) or serial hermaphrodites (e.g clown fish).
It's also politically useful - it enables us to say "these features of our sex make us vulnerable to male violence - lesser physical strength, vulnerability to rape and enforced pregnancy, having children dependent on us for long periods of time." It enables us to say "but biology isn't destiny - these biological differences may explain how come we are easy to exploit, but it doesn't say we have to be exploited. We can campaign for a better world." It enables us to say "having a uterus doesn't mean less of a person, of less moral worth, of less political worth, of less intellectual worth and so on."
Contrast this with "women are whoever says they're a woman" or "women are people who like pink glittery shit." Crap as a means of distinguishing the underlying biology, crap as a definition for scientific purposes, and deliberately crap in political terms, because if you can't name your oppression, you can't fight against it.