Lwaxana Troi is desribed on [TV Tropes](http://tvtropes.org/pmwiki/pmwiki.php/Series/StarTrekTheNextGeneration) as a "Betazoid Ambassador to the Federation".  On the [Memory Alpha](http://memory-alpha.wikia.com/wiki/Lwaxana_Troi) 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?