In English, "America" is the correct short for for the United States of America, and "Americans" for the residents. That's not true in other languages languages (Spanish, for example). I assume your moniker is less than accurate as in German "Amerika" and "Amerikaner(in)" are standard terms as well.
Yes, in English: America = US. In Spanish: America = The Americas.
I didn't knew that, thanks a lot for the info.
Edit: just another question.
What about American like saying European?
There's a similar example in Europe, but much more benign and regarding the water between England and France. In English it is known as the English Channel. In French, it is known as The Channel.
Well, if you were from the country most commonly known as "Mexico" you would really be from "The United States of Mexico". Therefore it makes more sense to call you a Mexican than an American.