I wish american english had a simple word for someone from the US. In Spanish they have estadounidense. I would prefer to call anyone from Alaska to Ushuaia an American and call people from the US a different word.
The common English word the country is "America". The continents are "The Americas" or separately as "North America" and "South America". The word for a person from the US is "American".
You might be interested to know that アメリカ (the romanization of America) is also how the country is referred to in Japanese. There are many other languages that followed English's use of America as the common name for the United States of America.
A notable exception, and where much of the disagreement comes from is Spanish. I generally try to follow the convention of the language I'm speaking and would never try to push the English-language conventions onto Spanish or vice versa.
