Pointe is a sort of spin-off of ballet. The first pointe shoes were just ballet slippers with the toes darned so they were stiffer. Loads of people know this, and so they link ballet with pointe automatically.
I have a simmilar problem. People call Irish dancing "Riverdancing" and that isn't right. Irish dance is a kind of dance. Riverdance is a show. Do people do "Nutcrackering?" No, they do ballet. I do Irish dance, not Riverdancing.
Well as you probs already know you can be in regular slippers or on those fancey painful pointshoes. When they start danceing on pointe they call ballet pointe as a sslang term. Basically saying I do ballet and Im on pointe.
Example: "Hey Brigette, sorry I cant comeover I have pointe after school."