Yes, people like me who would kill to just be recognized as women, the whole fight against being perceived as male, yes we are trying to take the term we want applied to us away (while they once again act like trans men don't exist) makes sense.
Another war on Christmas, that's what this.
As for the medical stuff while the terms here are for trans men, they ignore that since trans men existing defeats a lot of their arguments, but it works for trans women too.
For instance I am middle aged and just transitioning now, so there are male part exams I have to get while living and transitioning into a woman. Do you think when I have to go get a prostate exam I want to be misgendered or treated like a man? The point of the language is purely medical and due to medicine and tech letting trans people change their bodies, leading to men with biologically female parts, and women with biologically male parts. The language is so we get respect while also getting more accurate healthcare, something we are woefully underserved on as is and gets worse in a world that calls a Pap smear "a well women check". Remove gendered language for technical medical care is what we want, not "take away the term woman and woman's rights to call themselves women."
News flash, I call myself a woman too.