Ring species are so called because their breeding relationships form a cycle. You're only describing a chain. You would have described a ring species if you
could mate with your dog.
(Also note that while it's perfectly possible to walk up from humans to an ancestral slime creature, and then back down to dogs, most of the steps along that path will be shared between humans and dogs, like a Y with a very, very long leg. The recent common ancestor of humans and dogs would have already been a mammal.)
