If you are from the UK, you should probably clarify what you mean by a "Public school" since a reader from the US probably interprets that differently than what you mean.

Well its simple. I, personally went to two public schools, both of which were in the private school sector.

No, it isn't obvious. In the UK the prefix public and private are both synonymous for "fee paying" for schools.

I went to a prep. school (9 to 13) and a public school (13+) after that. Both of which would be described as private schools.

Yup. A huge number of rich foreigners send their children to UK fee paying schools. Lots of Middle-Eastern, Russian and Chinese. Something like 15% of current world leaders were educated in the UK.

Certainly caused some mumblings about the general character of some of the most prestigious institutions like Eton. It would be easy to put that simply down to xenophobia but there is a relative difference between the established culture and the new. British aristocratic elites are generally (and of course this is all relative) far poorer than the children of Oligarchs and party officials. Generally the attitude of the aristocratic Brits is to keep their head down and not be flashy, whereas some of their classmates literally have a private jet for their use on weekends.