Charles Lindsey wrote:

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).

There isn't a single universal definition and it's confused further by a lot
of the schools in question seemingly going out of their way to avoid using
the term and instead declaring they are an "independent school", which is
really synonomous with "private school". Not all private schools are in the
HMC, which also contains some non private schools (e.g. the London Oratory).
And not all are registered charities - a significant number (including my
old prep school) are run as businesses by Cognita.

A "public school" is generally one that teaches from about 13 upwards, with
the earlier tiers being "preparatory school" (or "prep school") from about
c7-13 (called prepatory because they are preparing pupils for the entrance
exams) and "pre-prep" from c3-7. Note this contrasts with the US where "prep
schools" are for university preparation (a specific concept that doesn't
really exist in the UK) and in their class system ex prep school pupils
occupy a similar place to "public school old boys".

However not all c13 upwards private schools are considered "public
schools" - it's very rare to hear the term used for any all-girls school.
And quite a lot of people will argue that a particular school is or isn't a
"public school" on the basis of one list or another, usually from the 19th
century (thus excluding all 20th and 21st century establishments) such as
the Clarendon Commission or the Public Schools Yearbook.