The word "ambassador" has specific legal meaning. From the Merriam-Webster website:

"ambassador
1 : an official envoy; especially : a diplomatic agent of the highest rank accredited to a foreign government or sovereign as the resident representative of his own government or sovereign or appointed for a special and often temporary diplomatic assignment

2 a : an authorized representative or messenger b : an unofficial representative <traveling abroad as ambassadors of goodwill>"

"representative:
1 : serving to represent

2 a : standing or acting for another especially through delegated authority

b : of, based on, or constituting a government in which the many are represented by persons chosen from among them usually by election

3 : serving as a typical or characteristic example

4 : of or relating to representation or representationalism" Jimmy Carter would fall into the 2b definition of ambassador above, as could ordinary Americans overseas. More accurately, ordinary citizens would fall under definitions 1 and 3 or representatives as you described their roles. True ambassadors enjoy perks like diplomatic immunity that simply don't apply to ordinary people. US citizens and military personnel are subject to being arrested - ambassadors aren't. True ambassadors from the US are appointed by the President and confirmed by the Senate - ordinary citizens aren't.