In any reasonable person's mind, a democratic republic (as seen by internet libertarians and conservatives) is a subset of democracy. Same with the democratic monarchies of Northern Europe, and the (more) direct democracy of Switzerland. Splitting them up just confuses people, and makes your point harder to communicate and understand.
never understood why conservative and libertarians americans want to make a difference between republic and democracy, it is literally in the definition of "republic" that it should be a democracy
Because it lets them make the implication that the Democratic party is un-American. And also that somehow the language is unfairly biased against them.
It's absurd, just like the bizarre fetish for using the term "Democrat party" as some kind of slur. But that's the essence of American politics now: a constant culture war, where the primary thing people want out of elections is to hurt the other side rather than pursue actual political goals.