Not quite. Harrow, Eton and Winchester are public schools but somewhere like Bromley High School and most other no-name fee-paying schools are called independen

Not quite. Harrow, Eton and Winchester are public schools but somewhere like Bromley High School and most other no-name fee-paying schools are called independent or private schools. Nobody would refer to one of these as a public school. A school that you don't pay for is usually referred to as a state school.