RE: What is a woman?
The word "Woman" has already been redefined to include transgender women. Your subconscious has already decided for you that transgender women are women, otherwise you wouldn't immediately understand what everybody is talking about when they say those words.
English is a living language that evolves over time, unlike Latin, where the meaning of "Cisgender" actually is set in stone, and therefore you don't get to decide whether you like that word or not, it simply refers to you whether you like it or not.
The meanings of English words are arrived at by cultural consensus, and the consensus has already been decided without any need for your approval.
Transgender women are women. You can tell that this is already true by the fact that your own brain had absolutely no confusion about what I was referring to when I said "transgender women." I meant somebody like me, not somebody like Elliot Page. That's how language works. One person says something, you understand what it means, therefore it means that thing.
Case closed.
Thank you for coming to my "Introduction to Basic Etymology" lecture.