I'm gonna answer this post as a trans woman myself.
I think concern trolling about the meaning of words has a transphobic flavor because in this particular context this rhetoric is used to take away validity from our identity. It's used to label us as second-class women, seen in your complaint about us claiming the term "women" for ourselves, which you seem to take issue with.
Here is the deal. Trans women don't take anything away from cis-women by identifying as women. And I think it's very important to internalize this. You don't own the word and we don't need your approval to exist, how about that? And look, let's get real for a second. It wasn't us trans people that took away your right to abortion in the US, right? Trans women were on the front line for womens rights ever since we have existed. We have been on your side when it came to questions regarding bodily autonomy for decades. I'll tell you why, the freedom to make autonomous decisions regarding your own body also affects us because transitioning is a severe intervention into the biology of our body.
So this entire discussion is quite odd, to be honest. You need to point the finger at cis-men because this society in many aspects still has patriachal structures. They've been oppressing you for centuries. And when they take away your rights and point the finger at us, it's a strategic distraction and it seems you fell for it. You need to point the finger at the people who are in charge of your rights and that's not us. The same people who took away your right to abortion also wage a war at our right to transition right now in the US. Nearly
240 anti-LGBT bills have been filed in 2022 alone and most of them target trans people. We're your allies, not your enemies.
And: other cis-women who oppose womens rights (also known as Republicans) have more structural power to take away your rights than we do and they're very often complicit in the conservative agenda to roll back your civil rights. It doesn't matter how you spin it, we've never been the reason for your misery.
This talking point has been debunked thoroughly among academic circles. Here are some scientific papers that researched the claim regarding predators and creeps invading womens spaces for example as a result of self-ID laws, something that has existed in Switzerland for
almost 8 months now. I've uploaded the highlights of said studies for you, in case you don't have the time to read through them.
https://onlinelibrary.wiley.com/doi/full/10.1111/1468-2230.12507
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https://research-information.bris.ac.uk/ws/files/139271435/Bristol_Pure_Version_PD.pdf
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https://escholarship.org/uc/item/4rs4n6h0
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Long story short; the idea that both trans-women and cis-men would invade womens spaces as a result of trans-affirming laws is fear-mongering that's based on anything but facts and repeating that argument doesn't contribute anything constructive to the discussion, the opposite is the case - you just reinforce the notion that trans-affirming laws hurt cis-women when that's not the case.
By the way, here in Switzerland we didn't experience any abuse either ever since we have implemented self-identification laws, which means you can change your gender through a very simple procedure that doesn't require a psychatric evaluation anymore. And I support this change because it means we're moving forward as a progressive society.