Here I think the teacher made a huge mistake, calling you both out in front of the class. They know how kids can be and an event like this can trigger bullying easily. This case should have been dealt with privately.
As about public and private school, in my country almost everyone goes to public school. Most likely there are private schools as well, but in 99% of the cases, that is out of the question, there's no need for that.
RE: How I remenber school ❤️🎭 - Como recuerdo la escuela ❤️🎭