Anastasia is correct. However, rector is used more frequently as a title for a church official -something like a vicar.

I would say Principal and Headmaster can be interchangeable, depending on the school but not limited to public vs private. Schoolmaster just sounds VERY archaic [ie, old-fashioned], as does rector. Principal is modern and commonly understood.

Hi! A principal is “a chief or head, particularly of a school.” "Principal" can also be used as an adjective meaning “first or highest in rank, importance, or value,” as in The principal objective of this article. "Headmaster" is a man heading the staff of a private school. "Schoolmaster" is simply a man who teaches children in a school. "Rector" is a senior official in an educational institution, and can refer to an official in either a university or a secondary school. Hope this helps :)