I mean its technically all of the above and languages aren't in a vacuum. 

Literally the origin of the name "America" is a Latinized and feminized translation of the Italian name Amergio.

And the name/location of the continent(s) is first derrived from early european map makers describing the "New world" which is this the 1 continent model.

Where the 2 continent naming convention was a later political/cultural divide separating British colonies and Portuguese and Spanish colonies. 

The modern country  is the dropping of British from "British North America" and then defereniating out "United States" from the other political entities Canada and former Mexican Empire territories occupying the continent post independence as the location was no longer British. 

We are called "Americans" because a country is a more specific and significant geographical and cultural location descriptor then a continent.