America... but which one?

There is the common habit of saying, as U.S. citizen or as a person living in the U.S., that "I am an American;" or "I live in America." What is vital to recognize is that there are four Americas: North America, Latin America, Central America, and South America.
I had a mind-opening experience that made me better conscious of this fact when I was studying in Spain during the end of my high school year. Not remembering how to say I am "estadounidense" [from the U.S.] when someone asked me what my home country was, I replied "soy de america" [I am from America]. That person then looked at me with a confused expression and told me that she and her classmates were taught in school that there is more than just one America. She explained how they addressed Canada as a part of America [being in North America] and then went on to mention Mexico [having some of their states in North America and the rest being referred to as a part of Latin America], etc... I then realized the utter truth to her response.
Yes. I still feel incredibly ignorant for not being conscious of the implications of saying that I was from America -- those implications being complicated and not specific.
Ever since my time in Cuernavaca, Mexico, this summer I have been asking myself "What makes people from the U.S. more entitled to claim the entirety of America [or, all of the Americas, for that matter] as their own?"
I realize this sounds harsh, but I feel that the truth in this is harsh and it needs to be more consciously recognized.
I do also want to add, though, that there are a great deal of people who use the term "America" loosely because it is what they have grown up around, because it is a habit, not meaning to take away of the significance and presence of the other "Americas"... I would just like to raise awareness of the implications and vastness of the term.

Thank you.