I always thought it was presumptuous to say that only US citizens are “the” Americans, since “The Americas” is a pretty broad term encompassing North and South America.
However, this being said, I think generally speaking in terms of nationality, “American” is thought to be a citizen of the United States, or one seeking permanent residency in the US. I’ve always thought this term was troubling, since I have a Colombian friend who was asked how long his family has been “in America” by a man from Spain. Latin America was not distinguished in the question. So in terms of culture, “American” is a troubling concept due to melting pot dynamics, as a solid ethnic identity is not assumed, as is the case in Asia or many European countries.