T1 has no previous experience with DID, but he talks to whoever wants to talk. He will sometimes ask "who said that?'. T2 talks to parts but rarely calls them by name. T3 and I identify which part she will talk to in a session, but she does not call them by name. Ex-T talked to them by name and called them out by name. T2, 3 and Ex all have experience with DID.
I think it is a style thing. Since you are your t's first client with DID, I think it would be ok to discuss it with her and see what works best for you. Some ts really want to map the system and figure out how it all falls together, some focus on what is happening day to day and treat the symptoms that are presenting.