I do not think there is enough information about the man in your story to determine his authentic self.

In your example-
I think controlling can be part of personality without being core self. It would seem something like controlling would be to protect the core self. Whether it is something that needs changing, is up to the person - ie. does it interfere with something you want/a way you want to be in the world and if so is changing it worth the effort/loss for you to do so?

Do you remember the part in Becket by Jean Anouilh where Henry complains that Becket is always defining things and Becket responds by saying without definition the world has no shape? I believe that. I can identify with your desire for definition.

Is the fact that I love these sorts of discussions part of my authentic self? Possibly, but I would more say it is more a part of my personality. I have different identities which direct my personality depending upon whether I am being professor, lawyer, friend, lover etc. And although my personality is somewhat the same - it manifests somewhat differently in each situation.