I hope you realize there's a fuzzy line between those, when you have ideal test doubles, that behave like the real thing, but are just configurable in-memory replicas of the dependency.

People get suck a kick out of categorizing their tests, I honestly find that funny.

Can you run it easily before you commit code to a unit, without a dedicated setup? Yes, then it's a unit test to me.

If it was me I wouldn't call it "something" test at all. It's just automated tests. The rest is mostly BS.