> If someone with limited English asks where I'm from, they're more likely to understand if I say "America" than "United States". And no one has ever asked "oh but which country in America?".

That's interesting! In my experience, meeting people from the US outside of the US mostly, when you ask where they are from, they tend to always say the city or the state, but never actually specify either "United States" or "America". If you don't happen to know that that state/city is in the US, and they never specified the country, you're usually safe to assume they're from the US :)