I get that some people, or many people, or some book, have defined it that way, but others have used it in the meaning as distinguished from plain refcounting. Many developers have always understood GC and refcounting (in languages with mutation) to be mutually exclusive concepts. That definition is used in practice, and when people say, "I'm going to implement garbage collection for my programming language," what they mean is one which doesn't leak memory.
A reason people use this other definition may be that plain reference counting and "having a garbage collector" are mutually disjoint, even if you define "garbage collection" as including reference-counted implementations. Making garbage collection a distinct concept from the set of acts a garbage collector might do will always be fighting against how language usually works.