I think the definitions and nominations of schools are different kn various countries. These terms can be specific, local titles in our country which don't have any exact counterpart in another country.
Anyway, let me write something concise about school types only in the USA. Of course, I'm not fond of this country at all, and this is just an example.
1. Nation's Public Schools:
Public schools get their financing from local, state, and federal government funds. Charter schools and magnet schools are two relatively new kinds of public schools.
1.1. Charter Schools:
A charter school is a school which receives public funding but operates independently. Charter schools do not charge tuition.
1.2. Magnet schools:
They are free public schools that can be highly competitive and highly selective. They're renowned for their special programs and high academic standards. They may specialize in a particular area, such as science or the arts. Students who apply to these schools may go through a rigorous testing and application process.
2. Private schools:
Private schools rely on tuition payments and funds from nonpublic sources such as religious organizations, endowments, grants, and charitable donations.
2.1. Independent Schools
2.2. Parochial Schools