Controller is usually reserved as the implementation that satisfies an API, regardless of what it does. Operators hide operational complexity via judicious use of APIs - if using it in the broadest sense the ingress API + a particular controller implementation (nginx which is a deployed object vs an ALB which might just be programmed) is an “operator”.

A Postgres operator could manage Postgres instances in pods, or it could program AWS RDS. Having a deployable entity isn’t really required. The thing reading the API and updating those objects is definitely a controller.