Meaning – What Does ‘That Public Schoolboy Polish’ Mean?

I would like to know what "that public schoolboy polish" means in the following sentences:

‘Yup,’ Will says. ‘We were at school together.’ I’m surprised. The
other men have that public schoolboy polish, while Johnno seems
rougher – no cut-glass accent.

‘Trevellyan’s,’ Femi says. ‘It was like that book with all the boys on
a desert island together, killing each other, oh Christ, what’s it
called—’

‘Lord of the Flies,’ Charlie says, the faintest trace of superiority
in his tone. I might have gone to state school, it says, but I’m
better read than you.

• Lucy Foley, The Guest List, Chapter 12

This is a thriller novel published in 2020 in the United Kingdom. One hundred and fifty guests would be gathering at some remote and deserted fictional islet called Inis an Amplóra off the coast of the island of Ireland to celebrate the wedding between Jules (a self-made woman running an online magazine called The Download) and Will (a celebrity appearing in a TV show program called Survive the Night). The day before the actual wedding day, Hannah, the wife of Charlie (Jules' friend), arrived at the island and is now at the dinner party for the rehearsal dinner with only some selected guests. And during the party, Hannah hears how almost all the ushers went to the same school as Will.

In this part, I wonder what "public schoolboy polish" means.
Does it perhaps mean that they have a sheen, a particular shine that would emanate from public schoolboys…?

But then, this is my small question, but is it right to understand that "public school" here actually means exclusive private schools in the UK, like Eton? This might be a fundamental question, but I am just wondering why it is "public" school when it is actually a "private" school.