As you’ve put it this way, I’d like to drill down into this a bit.
“Republic” is like the Family level of political taxonomy. Kind of like “Bear” in the animal hierarchical taxonomy. If someone says it’s a bear, but not a grizzly, well then, what species or sub-species is it? Maybe a spectacled bear?
The ‘species’ under Republic are: democracy, oligarchy, and autocracy. Right-wingers have a tendency to reference the Republic to indicate that they mean NOT (horror of horrors) democracy – or they’ll just outright say 'we’re a republic, not a democracy – but that begs the question: are they saying we’re an oligarchy or an autocracy?
So, if you think of the U.S. as being a republic (higher level classification) but not a democracy (sub classification), then what kind of republic are we?