I think some other countries historically used British terminology and maybe some of its own (e.g. "government school").

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).