Peter Masson wrote:

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.

Though many of the top independent girls' day schools come under the aegis
of the Girls Public Day Schools Trust.

....which dropped "Public" from its title back in 1998. Most of the GDST
schools have a much broader age range from about 3 to 18, reflecting
different arrangements in the girls' private sector (I think these were
largely driven by the GDST itself). "Public Day School" is a phrase that
seems to be almost exclusively linked to the GDST and of course these are
day schools whereas public schools have traditionally been associated with
boarding. It all just adds to the chaos and confusion.