People who call Queen Elizabeth II "Betty Windsor" are either anti-monarchy, small-r republican, apathetic, or satirists.
American: Dude, I just go to school here. I'm not Canadian. Why should I care anyway? Betty Windsor's just a person, like anyone else.
English monarchist: How dare you sully our most Serene and Exalted Monarch of the United Kingdom and the Commonwealth Realms with a made-up commoner name!
American: Let's get real. She eats out of Tupperware every morning at breakfast.