No they're not. Reference counting is a garbage collection algorithm. Its even described as such in most literature on garbage collection.

Reference counting and garbage collection are different, yes. But in terms of classifying "no garbage collection" under "things that Swift took from Rust", Swift's pervasive reference counting is much closer to classic tracing GC than it is to Rust's system of single ownership.

Automatic memory management is a synonym for garbage collection. Reference counting is a form of garbage collection.