...in a political science sense this is literally correct: democracy is direct rule by the people and a republic is rule by representatives. Further, in classical political terms, a republic is a "good" regime and democracy is a "bad" one.

However, and it's a big however, virtually no one uses the terms that way outside of political science wonkville.

It's basically just folderal that conservative academics like to throw out occasionally to show how overeducated they are. In any case, the United States is technically a democratic republic: we have a representative government, and the representatives are chosen by popular election. There are historical examples of other types of republics, which were not democratic, the most obvious being the Roman Republic.

The short of this is that our government is both democratic and republican, in theory at least.