paddyann, this is an England/Scotland divide. In Scotland the public schools, such as the one some of my DGC attend, which has XX Public School on the stonework is, in fact, a school open to the public, and a fee-paying school is a private school.
There is also a difference in secondary schools - when comprehensive education came in, schools in Scotland kept their old names - X Grammar School, X Academy, X High School, etc. This didn't happen so much in England. Our DC went to a local comprehensive which was called X Grammar School as that had always been its name.