Don't believe all what your teacher told you
It seems to depent from which point of view you look at the question, but my explanation for the olynpic flag seems to be more apropriate
The Americas (sometimes referred to as America) is the area including the land mass located between the Pacific Ocean and the Atlantic Ocean, generally divided into North America and South America. The term also usually includes the Caribbean, the islands in and around the Caribbean Sea, and Greenland, though not Iceland, for cultural and historical reasons. The isthmus of Central America is usually considered geographically part of North America. The Americas are often also described collectively as the Western Hemisphere or the New World.
Most references in English assume that that there are two continents, North America and South America. In American Spanish, however, the assumption is that there is a single continent, America. Moreover, the use of America to refer to the New World as a whole is also found in English, such as in the common phrase “Christopher Columbus discovered America”.
The single-continent concept also appears thematically; for example, the five rings of the Olympic flag represent the habitable continents; only one of the five represents all of the Americas.
People who live in the Americas are sometimes referred to as being American, although the word ‘American’ is used much more commonly, and, indeed, nearly exclusively in English, to refer to a citizen of the United States of America. The Spanish language uses norteamericano ("North American") or estadounidense (literally "United Statesian") when referring to U.S. citizens, and the French language which sometimes accepts ''
tats-unien'' ( tats-unienne for women). In Portuguese, people born in United States of America are mostly termed norteamericano instead of americano, while estadunidense is rarely used, and almost exclusively as an ideological statement that the term American shouldn't be reserved for the people of the USA, but many consider this should be the preferred name to be used. On the other hand, Mexico is properly the United States of Mexico.