From my experience running/overseeing groups (both furry and otherwise), I’ve used the term “zookeeper” to describe my work. I’m there to make sure everyone gets along, clean up the occasional mess, and if someone cannot play well with everyone else, ask them to leave.
The first and the second one sometimes mean counseling a member who is behaving out of character because they’re dealing with personal life issues, going through a rough time, etc.
That last item — asking someone to leave is the least desirable option, but sometimes sadly necessary. It can come into play when dealing with outright trolls, or to people whose mere presence is disruptive, because their actions elsewhere precede them.