Actually, this is merely a question about British terminology.
I became confused by this story in the Guardian:
The row over universities’ alleged discrimination against middle-class students intensified yesterday as private schools announced a boycott of Bristol University amid signs of growing disagreement among ministers about their own higher education policies.
The way we’ve always heard it over here, what Americans call “private schools,” the Brits call “public schools.” So what are the “private schools” mentioned in this story?