As a European, I don’t know what a Public school means. But if it refers to the kind of school environment you can find on YouTube by searching for “gen alpha teachers,” then I’m surprised those schools still exist at all.
It means "school paid for by taxes", which, as a European is probably the sort of school you attended
Private school is a school you have to pay to attend
The distinction between public and and private that British English historically had was useful too. State school is more accurate too (its distinguishing characteristic is state control and funding).
I also think the old (probably a century ago, or decades ago) "privately education" is a far more accurate description of what is now called "home education" in the UK and, even less accurately, "home schooling" in the US).