> So is it really not the case that "practice" is US English and "practise" is UK English?

Correct: practice is a noun, and practise is a verb, in non-US English. I don't have my (twenty-volume) copy of the Oxford English Dictionary to hand, but Wiktionary has an explanation under 'usage notes': https://en.wiktionary.org/wiki/practise

> British, Australian, Canadian, New Zealand and South African English spelling distinguishes between practice (a noun) and practise (a verb), analogously with advice and advise though without an analogous difference in pronunciation. In American English, the spelling practice is commonly used for both noun and verb.

I see.