We have a working definition, slippery though it is. It's your internal sense of self, in terms of being a man or woman. That suffices for social interaction, and differentiates enough from being a butch gal or a girly guy.
If that definition sufficed for social interaction, there'd be no call for preferred pronouns. An internal sense can't suffice for social interaction, because nobody else can perceive your internal senses. In order to interact with others in society, you need to account for how they perceive you. Traditionally, this was done by making a noticeable effort to "pass" as the opposite gender. But this was back when gender was synonymous with sex, and society's gendered expectations about dress and behavior were much more rigid and well-defined.
In modern times, it's done by aping traditional gendered dress, and by outright demanding (announcing preferred pronouns, etc.) that people behave as though they perceive you as the opposite sex.
All of which is effectively meaningless anyway, since in modern times there is no overt nor expected difference in the way we interact socially with each gender. We still treat the sexes differently in some cases, but gender decoupled from sex is functionally meaningless for social interaction. For example, people don't flirt based on the other person's internal sense of gender. They do it based on their own perception of the other's biological sex.
A much more realistic definition, which does not rely on invisible internal senses, and which is decidedly un-slippery, and which has practical applications beyond social interaction, is "adult human female". The only slipperiness to it, which baffles alien space robots but is immediately accessible to beings that communicate via natural languages, is that in some contexts the definition encompasses all human females, not just the adults (e.g., women's restrooms).
For public policy, though, it doesn't suffice. It gets interchanged with bio sex too capriciously.
Nothing capricious about it. Gender and sex have always been interlinked concepts in our society. Gender decoupled from sex has no practical applications. Your actual complaint, based on your recent arguments is that
we're not capricious enough in decoupling gender from sex (e.g., again, women's restrooms; the sex-based definition, applied consistently, handles this case just fine, but
you want to capriciously be inconsistent in this one case).
So for policy purposes, we need a more hard lined distinction. I think the hard line distinction will end up meaning 'it don't matter what you think of yourself as in the showers, but it do matter as the basis for interpersonal hate crimes.
We already have a hard line distinction: Adult human female. According to this definition, it doesn't matter what you think of yourself as in the showers, but it does matter that you're showering in the sex-segregated facility appropriate to your sex.
As for policy: We already have very clear policy that you cannot treat someone differently in housing, employment, etc. because of their outward gender presentation.
And note that this definition does not assail the dignity of trans-identified people. It makes no reference to, nor cares one whit about, allegations of mental illness or perversion. This entire discussion can be had - has been had, several times - dispassionately, respecting all internal senses and outward presentations, honoring the entire spectrum of human expression.