It’s a historical thing.

In the olden days the only way to get an education was through privately tutoring.

So establishments where everyone was educated together were “public” - even though you paid to attend.

Nowadays “public meaning paid” implies one of these original public schools (Harrow, Eton and the like).

I went to a paid school, founded in 1513, but it wouldn’t be called “public” because it’s not one of those original schools and therefore not high enough in the class system. It would be called “private”.

“Public as in free to attend” schools are known as state schools.

Just on a detail, I discovered recently that there have been various official definitions of "public school" and lists of public schools, and FWIW your school probably counts as one according to those. But I agree with your main points and your explanation.

The public schools would say their role is to prepare the future rulers of the country. And my school produced two former Chancellors of the Exchequer so I guess that makes sense.

Mine wasn’t a boarding school though which probably counts for something.

"State" school are the free, government run schools. "Public" schools are typically fee-paying, and are named so because when they were founded, anyone who paid could send their children there, rather than it being restricted by class or religion. They were "public" because the general public could send their kids there.

The ones everyone else goes to are generally referred to as "state" schools.