Yes, if you look up Democratic Republic in Websters Dictionary you will not find it, i doesn't exits. The only reference in the Democratic Republic of Yemen.
If you look it up on Wiki- you will find some nonsense about it being a hybrid of a republic and a democracy but all sources cited, which are only 2 are from 1997 and 2011. It is a made up term, and has nothing to do with the founding. You will find NOTHING in the writings of the founders referring to the US as anything other than a REPUBLIC. Not a Democracy, and not a Democratic Republic.
I would be more than happy to take away your ability to vote
We have been a democratic republic from the beginning, back to school with you.
Our founders knew too well the danger of a pure democracy, the same as with a pure republic. The examples in history are vivid. Most of the quotes you find are them arguing with those that wanted a pure democracy. The balance between the two is not perfect, but safer than either.