I think philips is exactly right. The whole design of Kubernetes is geared toward allowing users write their own controllers for advanced use cases. You could view something like Jenkins or Vitess as controllers because they spawn Kubernetes pods on demand. The beauty is Kubernetes gives you great primitives, so you often will be controlling these objects, not the underlying pods. Of course many simple applications don't need a controller, although I suspect more and more simple use cases will be managed by an external controller like Helm which orchestrates the lifecycle of applications.

The questions is: Why isn't this just called a controller? What's this new term Operator?