It's too bad we never got any dialogue describing Lwaxana Troi as either the Betazed Ambassador or the Betazoid Ambassador in her official quality. At least there would have been a clear grammatical clue there, unlike with Sarek being the Vulcan Ambassador...
From TNG
Manhunt
LWAXANA: Lwaxana Troi, daughter of the Fifth House, Holder of the Sacred Chalice of Rixx, Heir to the Holy Rings of Betazed.
DATA: Full ambassadorial status, sir.
RIKER: It looks like it's back into that dress uniform, sir.
DATA: And yours as well, Commander. She is listed as representing the Betazed government at the conference.
LWAXANA: I am be serving a Betazoid dinner of greeting tonight, Captain. It is an ambassadorial function.
PICARD: It sounds delightful.
PICARD: Gentlemen. I think we must not lose sight of the fact that we're talking about someone who has been granted ambassadorial rank. Even though she may appear somewhat eccentric, Lwaxana Troi must be treated with the appropriate respect. Is that understood? Picard to Pulaski.
Half a Life
LWAXANA: Well, I am sure! I am a Betazoid ambassador. I'm a Daughter of the Fifth House, and those people are going to answer to me! So you just energise this damned thing and get me down there!
From DS9
The Forsaken
LWAXANA: Oh! Where is it? It's gone! It's gone!
BASHIR: What's gone, Madam Ambassador?
LWAXANA: My latinum hairbrooch.
SISKO: Lwaxana Troi the Ambassador?
ODO: That's her. There was a minor incident at the bar that I helped her with, and now she's grateful.
Fascination
ODO: Madame Ambassador.
LWAXANA: Oh, do we have to be so formal?
ODO: So, what brings you to the station.
LWAXANA: Officially I'm here as the Betazoid representative to the Gratitude Festival. But the truth is, I came to see you, you poor sweet tortured man.
It might be possible to play with the semantics from Fascination, but Manhunt is clear she represented Betazed. It's also clear she's an ambassador; however, it looks like she was appointed ambassador sometime after TNG's first season.