Learning what it Really Means to be 'American'

At times in conversations here in Colombia, I will refer to being American and I have been told by more than one person, that people here are Americans too. The more accurate description would be to say, people from the United States. People from Central America and South America consider themselves American. People from the United States seem to think the word applies to them only, when in fact, it doesn't.
I realized that I have not heard any native Colombian refer to people from the States as American. They will rarely say America, they will say 'United States,' when referring to the country itself. 
I do know that in Europe and the Middle East and perhaps other parts of the world may not differentiate and say Americans when referring to US citizens or the country. However, here in South America, people are well aware that there many countries whose citizens are also American. Kind of puts the whole immigration discussion in a new light doesn't it. At least it does for me.