I'm sorry, but whatever elementary-school teacher taught you (and millions of kids like you) this stupid thing needs to be slapped. It's not that it's unnecessarily pedantic. It's just wrong.
Whether we're a "democracy" or a "republic" simply depends on whether you favor Ancient Greek or Latin.
The word "democracy" derives from the Greek terms "demos," meaning "people" and "kratia," meaning "power" or "governance." So a "demokratia" is literally a "government by the people."
The word "republic" comes from the Latin "res," meaning "thing" and "publicus" meaning "of the people" or "public." So the Respublica was the "thing of the people."
In other words, the two terms--democracy and republic--mean exactly the same thing in two different languages.
Somewhere people got confused, because Athens was a direct democracy and Rome was a representative democracy. And since Rome gave rise to the term "republic," they erroneously concluded that a "republic" only meant a representative democracy. And if a republic is a representative democracy, then for some reason a "democracy" can only be a direct democracy.
But that makes no sense, either linguistically, historically, or logically.
We are a republic. We are also a representative democracy. It's the same fucking thing.