I was actually talking about automatic reference counting. I think both Rc and Arc count as automatic reference counting? It's just that the term (which was apparently coined by Apple?) is not commonly used outside Apple's languages.
I'm not familiar with Swift though so my understanding could be incorrect.
My understanding is that Rust doesn't have _automatic_ reference counting as in Swift only because it has an alternative (move), which requires the programmer to specify their intent. The principle is nevertheless the same: ensure every time a reference is copied the ref count is incremented, free only when ref count is zero, and we get temporal memory safety.