This is the crux of the matter.
Why is it so very important to call yourself a woman if in doing so, you have stripped the word of all meaning?
The word "women" can either mean female people, or it can include male people. It can't do both.
If it includes both male and female people then it has the same meaning as the word "men" and there is no basis for determining who is in which group or even that there are two groups.
Obviously there are still two groups and always will be, but the trans activist position appears to be that we don't have or need words for those two groups and we certainly don't organise single "gender" spaces or sports according to them. So given that we currently (according to trans activists) organise these things according to two groups which are the same as each other, because they both now mean "some humans", surely that's an argument for not having separate spaces and sports for men and women at all. And yet, they're, not campaigning for that.
This shows that, paradoxically, the word "women" has to have meaning, and it has to mean female people, because otherwise there would be no "other" group for trans people to identify as. The point is not the abolition of rigid gender categories. The point is for some men to be allowed in women's spaces.
I don't care about people identifying as cats or bagels; in fact, I welcome it. Because a man can no more be a woman than a teenage girl can be a cat or a bagel. It's all just claiming to be something you are not, and I welcome the discussion about why it's ridiculous to claim to be another species or a baked good, but not ridiculous to claim to be a woman with a penis.