Write all the countries (enclaves included) you meet, starting on the west side of the Panama Canal and moving south, until you've gone all round America, and are back to the west side of Panama Canal
I realize I'm about to incur much wrath poured upon my head, but IMO, America refers to the United States of America. North America and South America refer to the continents, America is an abbreviated form of the country, just as someone from the Federated States of Micronesia would say they are a Micronesian, not a Federated, or someone from the People's Republic of China would say they are Chinese, not People. If someone says they are American, people usually assume they are from the USA. I've never heard a Canadian say they are Canadian-American, nor a Brazilian say they are Brazilian-American. If they did one might think they were born in the USA to Canadian or Brazilian parents. Canadians are North American and Brazilians are South American, just as the French say they are European or a Nigerian would say he is African. I honestly do not think it is arrogance when people from the US say they are American. It is a typical abbreviation for most world countries which have long names.
I'm not saying that the term America does not refer to United States but I assume you can also say that "North America + South America = America". After all South America certainly doesn't mean the South of the United States. No French would say he's a French-European, if he says he's French it's pretty obvious he's European, too.
Had a quick look on wikipedia and looks like while Americas is better, America is fine, too -> http://en.wikipedia.org/wiki/Americas
Had a quick look on wikipedia and looks like while Americas is better, America is fine, too -> http://en.wikipedia.org/wiki/Americas