Lwaxana Troi is desribed on TV Tropes as a "Betazoid Ambassador to the Federation". On the Memory Alpha page, she is named "a Betazoid Federation ambassador", and elsewhere that she "represented the government of Betazed".
I cannot recall any particular line of dialog from either TNG or DS9 that establishes exactly for whom and to whom she was an ambassador. Was she an ambassador of the Federation or of Betazed? Or both? Was she an ambassador to the Federation, Betazed, or other planet(s)?
If the answer is that, as TV Tropes seems to believe, she was the Betazoid ambassador to the Federation, then why would a member planet have an ambassador to their "national" government? Would this not be the equivalent of Florida having an ambassador to the United States?