Americans are justifiably proud of their democratic political institutions and often believe that people of all nations would benefit from living under American rule

Are you asking what the name for this idea is: It is called "American Exceptionalism": the belief that American democracy is special and that the political systems elsewhere are not as good and should be exchanged for the one in the USA.

As for whether this claim is true: it's relative. While democracy in general is connected to people's well-being, more than dictatorships, American democracy should always keep on evaluating its effectiveness. For example, some people argue that the Electoral system is not very effective and leads to people being chosen as a president despite not winning the mayority of the votes.