AITA for pointing out my girlfriend misspelled her son's name?

My (39M) girlfriend (36F) and I have been dating for 2 years. She has a son (5M) from a previous relationship; his dad has never been in the picture. Her son's name is Cary.

I was planning a trip to Europe for all of us and needed to start the process of getting Cary his first passport. I asked my girlfriend for his birth certificate, and she gave it to me. I was confused because the name was spelled “Carrie.” She said Cary was short for Carrie.

I explained to her that those are actually different names and that “Carrie” is a girl’s name. She argued at first, but I showed her she was wrong, and she eventually conceded it was a mistake. I suggested she should get it changed before he gets older.

She dismissed it, saying it’s essentially the same and that names for girls and boys are more flexible now. I insisted, and she got upset, accusing me of trying to make her feel dumb for the mistake.