I'm glad you were able to look it up.

There are a number of ways in which schools are funded in the UK. Either people pay themselves or the state pays. We tend to call schools which are funded by the state "State Schools" but never "Public Schools" as I believe is the case in the USA. There are a few grey areas in between, church schools in England and Wales are sometimes Voluntary Aided which means that the state pays some proportion and the church diocese pays some proportion. A school, attended by my children, was funded for 100% of the running costs but only 85% of the building costs - I'm sure there are other formulae. Some schools are known as “Independent Schools” or "Private Schools" where, like in many parts of the world the parents (or anyone else who can be persuaded) have to pay. Some schools have sums of money invested as endowments which help to fund the schools.

Within the independent or private school sector there is a subset of schools known as public schools. This is where your definition comes in useful because I would not like to say which schools consider themselves to be “Public Schools” and which are not. Almost certainly Eton (did you see Prince William in his tail coat?) and Harrow but also lots of others are called Public Schools. I suspect the reason is not just that they are the bastions of privilege. They are also historically some of the oldest schools, especially since in days gone by, poor children went up chimneys to earn their living and didn't get a chance to go to school. Therefore I think that the definition that they were the schools that opened their education to the public is correct.

I notice that the Independent Schools Information Service http://www.isis.org.uk/ is uses the term "so-called public schools", implying that it is a distinction they would prefer to see abolished.

I suppose the nearest term in common use in the US would be "Ivy League" Universities like Yale and Harvard compared the University of Western Smallesville.