When we were leafleting around the time of the GRA consultation in my home city (in England) some of the public thought:
A transwoman is a woman who wants to be a man, because of the use of the word woman in tw.
Before I started reading these boards, this is what I thought. When I heard "Transwomen are women" I assumed it meant women taking testosterone were still women and needed to use facilities like any other woman.
I knew about Gender Recognition Certificates but thought they were for transsexual people like Julia from the BBC2 documentary who was very dysphoric and had surgery on her genitalia.
I had no idea that it involved anyone outside that small group, or that it was possible to change all your identifying documents such as passports, without a formal change of sex.
I thought I was reasonably well informed. Who knew.
Since then, my DD has house shared with someone who has Asperger's and is transgender which seems to involve wearing skirts but reverting to male when DF picks them up. DS has gone out with a transgender person who went for full surgery and from a massively positive start has gone worryingly quiet. And my bestie's DD's bestie is now a boy called Kai.