I am watching an actor at work. Is it real? Are those "real" emotions? I don't care. You don't even have to be believable, just convincing. The difference is that, in the latter case, I the audience participate in the illusion. To convince me you have to take me through a process. It is a shared contract between the actor and me. Authenticity is in both our hands.
However, something that is so magical on the stage when it "really is real", is spontaneity.