Can you not see the difference between being prevented from calling yourself a woman and being called a cervix-haver?
To all practical intents and purposes, no. When I am being told that the word "woman" is not appropriate for these discussions, and its meaning has become so diluted that I cannot use it to define myself as an adult human female, that most definitely affects my use of it.
I can see the gnat's wing of difference you're scrabbling for here in that I can use the word for myself (why, thank you), but when the meaning is useless, and I'm not allowed to use it in a discussion about female biology, the difference is so subtle that it disappears. I can't call myself a woman and have it cover the fact that I'm biologically female.
I see you're claiming that it's actually about including transmen, and if you believe that, I've got a bridge to sell you.* If that were true, men's groups would be just as averse to using their word. Here's a snap I literally just took from the Prostate Cancer UK site (and to be clear, I absolutely support them using this language).
*It's actually a matchstick, but it still costs £5 million and will be used to traverse rivers. No problem with that, I hope? After all, London Bridge can still use the term.