Hi Bob
We do this deliberately. It's what makes us quaint.
Public schools are private, in that they choose their intake, and charge the parents hefty fees. They are like private schools, which do the same. Public schools sometimes issue scholarships for talented kids from poorer backgrounds, but those kids are not going to be in the same class. Well, the same class yes, but not the same social class.
There are also State schools, which are open to the public, though of course, they are not Public schools, and many members of the public can't get a place for their children. They either have to travel to another state schoold open to the public, further away, or pay for a private school or a Public one.
A form is a class. Not a social class, just a class, all the same year. We used to sit on forms in the school hall during assembly. So the 6th form is for those who stay on after compulsory education, to do A levels, often in preparation for University. As opposed to Prep schools which are in Preparation for Public schools (which, you will remember, are private). 6th forms are usually lower (during which year year they will usually have their 17th birthday) and upper (18 ).
Haven't come across Lower 3rd etc, we didn't have that. Have you been reading Billy Bunter? 3rd Years would be 14 years old, of course, (unless tey were 10) and maybe the Upper 3rd is the class of brighter children, but I'm guessing there.
Personally I went to an English grammar school. It wasn't a public school, anyone could go if they passed the exam, and we weren't taught grammar, except in Latin and French lessons.
I don't have kids, and having taught for four long dark miserable years inthe 80s, I come out in a rash just thinking about schools, but my understanding is that now things are different in that they are called Year 6 and Year 10 etc.
Hope that clears that up.
Cheers
Steve