The existence of DC and territories means it’s not a Republic either. The US is it’s own thing neither a union of states or any specific type of government, which seems oddly appropriate.
It appears that you misunderstand the definition of democracy, which is, a system of government where people may choose their leadership and/or laws, either directly or indirectly.
You can argue until you're blue in the face that the USA is not a democracy, but the definition or words is largely determined by consensus. And most Americans would disagree with you and refer to the American system of government as democracy due to their beliefs that voting is the most fundamental determinant of a democracy, and many Americans have first-hand experience voting for leadership, legislators, and laws.
Words don't have to be used literally. Everyone knew exactly what the original poster meant by democracy. Well-actually'ing on obvious things like this is just tiresome.
It's not a trivial distinction. A republic protects the _inalienable_ rights of the minority. It's not up to anyone. It's the reason a creator is valuable to even atheists.
