Er preSCHOOL!
Not sure which angle you're coming at this from - I would go for the opposite emphasis - PRE-school - as in its where you go before you start actual school! That's what the "pre-" prefix means, surely??
Re: labels - I don't think it's anything like as straightforward as some posters are suggesting. I would also at first glance assume 'playgroup' was the sort of thing that happens in a church hall and parents stay at, but I know of many (private) nurseries which use it as their name. Likewise I think there's huge overlap in the sorts of settings which call themselves nurseries or pre schools. I'm not sure whether 'nursery school' as a title requires that they have teachers on their staff? (All the ones I know do, but so do a lot of the nurseries I know - its outright untrue to say that "nurseries" are only staffed by nursery nurses.)
Likewise - being attached to a school doesn't imply one standard level of integration with the school proper, or carry all the same purported signifiers of 'school', as different experiences show. My son's nursery (with teachers!) involved no assemblies and no uniforms, and the staff were all addressed by their first names, unlike at the school it was attached to. The nursery class at his primary school wear the uniform (optional, but it looks like almost all do), but have limited involvement with the rest of the school. Most of the EYFS/KS1 teachers use first names, as do some of the KS2 ones. OTOH, I know schools locally where the nursery and reception classes function as a single unit.
I hope nobody would be dickish enough to correct a 3yo proudly telling them they go to school now, and of course a parent dropping a 6yo and a 3yo at the same place in the morning is being perfectly reasonable to describe that as the school run etc. But I would find it a bit odd tbh to see "first day at school" photos on fb of a child who has just joined the nursery class, even in a school, and if someone I knew talked about 'school' I would be a bit surprised if a year later they were talking about their kid starting reception etc. Because whilst I know people use the different terms 'nursery/ preschool/ daycare/ playgroup/ nursery school/ nursery class/ kindergarten/ Montessori (this one a bit more specific obvs)' to describe the institutions their PRE-school age children attend, I think it is broadly understood that 'starting school' in this country is something that happens at age 4-5 when they start reception class.
I'm not familiar with the practice of (some?) private schools starting 'school' at age three. I don't really interact socially with anyone who sends their child to private school so this doesn't cross my radar. I am saddened at the thought of three year olds following the reception curriculum tbh - it was a hard enough thing for my 4yo at a very gentle play-based reception class - but that's personal preference I guess. Agree with what others have said about 'selection' at such an age - horrible.