This involves no confusion, since there is no country in Europe, Africa or Asia were the name of a continent is used preferentially for its own inhabitant. America is the only continent where that has happened.

"America" is not a continent. The continents in the western hemisphere are North America and South America. There is no continent called "America." The continents considered together are referred to as "the Americas".

Does Central America count as a continent?

It is not separate from Mexico (which is universally considered to be in North America) by anything other than an imaginary line (national borders). It is a political entity. If Mexico had at some time in the past absorbed all the countries we now call "Central America", and extended down to what is now the border between Ecuador and Panama, the term "Central America" wouldn't exist at all.

In contrast, the boundary between European Russia and Asian Russia (aka Siberia) is the Ural Mountains, which form an actual physical boundary: therefore it is not a political boundary but a physical one. But it's not water so it's questionable whether the two are separate continents; hence the (relatively new) term, "Eurasia".