When the question is "what is a man?" or "what is a woman?" there is no one defining characteristic that is or can be conclusive of the issue. You are absolutely correct that every argument thrown at a trans person to exclude them from the gender with which they identify fails any logical test.
The only means in which it can properly be defined and which pass the test of logic include one which is internal or external. The internal means involves how one identifies in our heads. It actually passes the logic test.
The only other means, and this is external, is the duck test. If it walks like a duck, and looks like a duck and quacks like a duck, then its a duck. It is illogical for them to say "well that person over there walks like a woman, and looks like a woman, and dresses like a woman, and sounds like a woman, but I'll call him a man because when he was born he was assigned male". Gender critical people get very hung up on the issue, but in day to day life, when somebody is referring to me, they will naturally choose she, woman, her etc. Its not out of niceness (although they might be nice), its how language works. The shop assistant will say "can you help this woman here, she needs...." because its how descriptive language works. And the shop assistant gives not two jots about what genitals I may or may not have.
