Abigail Shirer and Deborah Soh - 2 authors, and you recommended 2 different books written by them.
We are talking about gender identity. Not sex. They are not the same.
It is one thing to say that a person identifies as, wants to live as and be treated as a person of the opposite sex.
This
"so what is a woman? can you enlighten us?
A person who defines themselves as a woman regardless of what their sex assigned at birth was"
is another thing all together. A woman is a woman, with an actual biological definition even if some scientists are backing away from reality in some Trumpian like desire to live a lie. If a man identifies as a woman, they are still a man. You can extend the courtesy of treating them as a woman, they can even feel as if they are female, but at the end of the day, it's a dude. A dude living as a female.
Nobody ever would question this with regards to someone who identifies as a cat [and they do exist]. Most sane and science grounded people would never say a cat is someone who identifies as a cat.
Also if a woman is someone who identifies as a woman, then there is 0 need for hormone treatments and sex change operations, because they are already fully women.