I am not certain what is considered a private school in USA, I think it is different in Britian to our system here, Private is where you pay through the nose for your education, for the wealthy really, but you get lots of extra priviliges and great schooling usually, Public is where my children attend , it is funded by the government, but we pay school fees etc, and there is nothing wrong with the education, as long as you are picky where you send them, of course we have zoning here too, so you can only attend schools within your area usually.

College in NZ is like High School, we have both High Schools and Colleges,university is where you go after finishing your high school years as long as you pass your exams.