In short, she was the ambassador to whoever she needed to represent Betazed to. From [Memory Alpha][1]:

> An ambassador was the official diplomatic representative of one government or political entity to another, or to an organization of political entities. Ambassadors also functioned as the heads of embassies. Customarily in the United Federation of Planets, ambassadors represented the Federation to non-member worlds, and sometimes member worlds had ambassadors to represent their worlds to the Federation itself. (TOS: "Journey to Babel") ....


  [1]: http://memory-alpha.wikia.com/wiki/Ambassador