I think the ship sailed on "America" meaning anything but the US quite a long time ago. Geologically, it's two different continents with different names plus some little bits of smaller tectonic plates. It's unfortunate, but there is no common culture, either historically or philosophically.

I think this issue really comes from government and education policy in Latin America. Unlike English, Spanish (and many other languages) have what amount to language police. This is the one that oversees Spanish:

http://en.wikipedia.org/wiki/Association_of_Spanish_Language_Academies

I'm sure that there's no criminal penalty but, for instance, a student in a Spanish-speaking country would be considered "wrong" to refer to the US as "America." It wouldn't just be a matter of opinion or preference. A scholar probably wouldn't be allowed to use the term in a Spanish language research paper. So there's really no way of knowing what common people in Latin America really think of "America" as the United States.

A couple of more things: Spanish speakers in the US are also governed by one of these entities:

http://en.wikipedia.org/wiki/North_American_Academy_of_the_Spanish_Language

And, these groups don't just say it's wrong to call Americans "Americans." It's also wrong to say that name of the country is the United States of America or USA.