Since the inception of the United States, has the term "America" ever referred to something more than simply "the United States"?

From a purely historical perspective, I believe the term "America" has generally been applied only to the "United States of America". I have seen instances where the collective countries of Mexico, US, and Canada have been referred to as "the Americas" (note the plural). However, I don't believe there has ever been an instance where the term "America" by itself has ever been used to refer specifically to any other country exclusively.

Latin Americans consider themselves "Americans," because South (and Central) America are part of the "American" continent.

To distinguish themselves from people from the United States of America, Latins call the latter NORTH Americans.

This could (but usually doesn't) refer to Canadians, who are generally referred to as "Canadians" by the Latins. Technically, Mexicans are "North" Americans, but again, this doesn't really apply.

To be honest, at first I was upset with your question. I mean, every child in Poland is taught that Christopher Columbus discovered America. How would it be possible if America would be precisely synonymous with United States, as it was written by you. Also my Brasilian friend Rodrigo, who stays at my place for few days, asked me to write it here that he defines himself as an American.

But indeed, I cannot deny that there are plenties of situations where people use words such as America or American speaking particularly about United States and Polish language dictionaries mention it as one of common use of those words or even the main one.

I've started to check it by myself. There's no mention of Americans in Declaration of Independence nor the Constitution. But I've found it in few first inaugural speeches of US Presidents.

1st inaugural speech by George Washington

(...) but not without resorting once more to the benign Parent of the Human Race in humble supplication that, since He has been pleased to favor the American people with opportunities for deliberating in perfect tranquillity (...)

2nd inaugural speech by George Washington

(...) When the occasion proper for it shall arrive, I shall endeavor to express the high sense I entertain of this distinguished honor, and of the confidence which has been reposed in me by the people of united America. (...)

Inaugural speech by John Adams

When it was first perceived, in early times, that no middle course for America remained between unlimited submission to a foreign legislature and a total independence of its claims (...)

(...) In this dangerous crisis the people of America were not abandoned by their usual good sense, presence of mind, resolution, or integrity. (...)

(...) Such is the amiable and interesting system of government (and such are some of the abuses to which it may be exposed) which the people of America have exhibited to the admiration and anxiety of the wise and virtuous of all nations (...)

(...) if, while the conscious honor and integrity of the people of America and the internal sentiment of their own power and energies must be preserved (...)

(...) if an unshaken confidence in the honor, spirit, and resources of the American people, on which I have so often hazarded my all and never been deceived (...)

(...) With this great example before me, with the sense and spirit, the faith and honor, the duty and interest, of the same American people pledged to support the Constitution of the United States (...)

Finally, the first mention of the word "American" as a synonym to citizen of United States, I've found in:

2nd inaugural speech by Thomas Jefferson

(...) The remaining revenue on the consumption of foreign articles is paid chiefly by those who can afford to add foreign luxuries to domestic comforts, being collected on our seaboard and frontiers only, and incorporated with the transactions of our mercantile citizens, it may be the pleasure and the pride of an American to ask, What farmer, what mechanic, what laborer ever sees a taxgatherer of the United States? (...)

I hope it will help you in your researches.

From a British perspective, America is not used exclusively as a synonym for the USA. I have several times encountered the conversation:

"Where are you from"

"America."

"Oh cool. Which country."

"The United States."

Another example would be a recent BBC program I watched talking about geography in Iceland. The presenter was standing on the fault line there and said, "My left foot is in America and my right in Europe." In this example America was distinctly not a synonym for the USA.

Yet another case, when describing the location of a country such as Canada or Mexico, a British person would describe them as being in America. Obviously not the USA but a greater America.

The plural form, mentioned in another answer, "Americas", is usually used to describe locations outside of North America such as Brazil, Jamaica, Columbia.