...confusingly known in the UK as "public schools"...

Agreed it's confusing. A little history (from memory, so it's probably wildly inaccurate):

Once upon a time (like in the Dark Ages), nobody went to school. Those kids lucky enough to have rich parents (nobles, etc.), had private tutors.

When the merchant class started to blossom, they wanted to educate their kids, too, but couldn't afford private tutors. This led to the foundation of schools that anyone could join, as long as they had the cash -- hence "public" school.

These days, most people use the two terms interchangeably.

Personally, I don't care -- I went to a governors' conference school .