Supervisor, Mentor, or Manager?

There are many academia-specific keywords: "grant", "stipend", "precarity". But what is the correct designation for the leader of a research group who writes projects, manages people, supervises students, etc? Manager? Supervisor? Mentor? Because these are very different concepts. It could be easier to stick with the concept of "principal investigator" (aka PI), I know. But, let's face it, that word ends up being a little bland right?