What is the history of the words Latin, Latino, etc, to represent all things Hispanic?
I think your confusion may come from thinking that English is a “Latin-based language”. The richness of modern English is due to its origins as a combination of Anglo-Saxon (i.e. Germanic), Norse and Latin roots, caused in turn by the history of England as the site of a wave of invasions from continental Europe, mostly from the 5th to 11th centuries. Latin America is therefore distinct from “English” America because of the presence “south of the border” of peoples speaking languages that have Latin as their major root, namely French, Spanish and Portuguese.