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[Kaedan]

Depends.
"America" is also used as a term for North America and South America together, and that was also the original meaning of the name.
Also, in many places America as a whole is actually considered to be a single continent. In other places it is considered to be composed of two continents - North and South America.

I'm an American. We call our country America. I'm from America. (not all that proud about it, evidenced by the fact I'm an expat in Japan, but that's besides the point).

What you and the other poster are referring to is "the Americas". That refers to the entirety of North and South America. No one from Canada, Mexico, or South America refers to themselves as an American or being from America. At the very best (for your position), they will say "I'm from the Americas". But most times they'll just say their country, North America, or South America.