"I don't need politicians to 'know what a woman is'; I need them to prevent people from discriminating against me just because I'm a woman."
They quite literally can't do that if the definition of woman is not adult human female.
For example, the Equality Act, designed to protect (actual) women against, for example, equal pay discrimination, is seriously hampered if the definition of "woman" is expanded beyond biological women. Broadly (because I am certainly not qualified to give great detail), in the case of equal pay discrimination cases, that is because of the use of 'comparators' in a legal sense to prove that the difference is because of sex, and therefore unlawful.
If the man or men (being paid more than the woman/women) who would ordinarily be the 'comparators' that would enable her to show sex discrimination are suddenly legally counted as women, not only can those men not be used as those opposite sex comparators needed, but their simple existence actually undermines her common-sense case because, look, that "woman" over there is being paid the same as the men, so the reason for the disparity in pay between all the men and the complaining woman can't sensibly be sex discrimination. That company is clearly perfectly happy to pay "women" the same as men, so there must be another reason for the low pay of the complaining woman.
Unfortunately the Scottish courts have decided that 'sex' in the Equality Act includes men with a gender recognition certificate so that's the situation we're currently in. That's in relation to ensuring that women are represented on public boards, but a man with a GRC will - according to the Inner House - do the trick. No biological women necessary. That's being appealed to the Supreme Court, but here we are. Sexist companies can in practice see the reality of sex and discriminate accordingly but the law is actively prevented from observing, acknowledging and compensating that reality.
Definitions matter a lot in practice. Don't be too quick to shrug about them being fudged or obfuscated.