Actually, it's worse than that! Fee charging schools can be known as both public and private schools. Those terms are never used in the UK for schools funded by the state. A state funded school will be labelled as Primary, Secondary etc. Some are known as State schools. Some are Academies (a bit more complicated but largely publically funded) etc. Basically in England anyway, Public and Private schools are fee charging schools. The name does not refer to how they are funded.

I went to a private school aged nine to 13 and a public school 13 to 18. Then I went to a polytechnic, which changed its name after a year and then two (three?) years later it was a university! Whilst in sixth form (17/18)

So the message here is that the public/private distinction for school nomenclature here in England and perhaps some or most if not all the UK doesn't mean what it does elsewhere, unless it does except where it doesn't ... except on a weekend when all bets are off. Clear? Jolly good. As you were, carry on!