In i Tom Anderson writes:

And, as you are in the process of explaining, hence the term 'public', as
opposed to the only other schools which existed before them, the grammar
schools (although i'm not sure if they were called that then), which had
entrance exams (and mostly still do).

Strictly speaking, I think the term "public school" is defined as

A private school whose headmaster (or now mistress) is a member of the
Headmasters' Conference.

There may also be a requirement/expectation that it is a not-for-profit
(and likely also a registered charity).

--
Charles H. Lindsey ---------At Home, doing my own thing------------------------
Tel: +44 161 436 6131 Web: http://www.cs.man.ac.uk/~chl
Email: Snail: 5 Clerewood Ave, CHEADLE, SK8 3JU, U.K.
PGP: 2C15F1A9 Fingerprint: 73 6D C2 51 93 A0 01 E7 65 E8 64 7E 14 A4 AB A5