[Sundae]

Ground floor is on the ground of course!

Private schools are generally called private schools now.
Very, very expensive and exclusive (and OLD) ones are called public schools.
The rest (such as the one I work in ) are called state schools.
But as it's the vast majority, so doesn't tend to be specified.

Private schools always used to be called public schools (because anyone with money could attend them), but not so much now.
But is someone is labelled a "public school boy" is will mean expensive education and usually family with old money and/ or land to back it up.