No; that would make us a direct democracy,

Okay, the simplest web page I was able to find to explain this was from an Australian site. I feel like the Kahn Academy site is perhaps a bit loaded, in that it classifies "representative democracy" as an "elite democracy", which it technically is, but it makes it sound bad, and the Wikipedia page gets bogged down in the weeds with a lot of stuff that isn't relevant to what we are saying, but feel free to check those out, too - but here's a simple page to summarize:
https://www.moadoph.gov.au/democracy/defining-democracy/

I am not arguing about the relative benefits of direct democracy versus representative republics. I am asking why you think that representative republics aren't a form of democracy - where that talking point comes from. Where did you first hear that talking point, and why do you repeat it?

"The United States is not a democracy; we are a republic" is a false statement.

"The United States is a democracy; we are a republic" is the true one.

"The United States is a democracy; we are a direct democracy" is also false.

I have never understood why people say this. A representative republic is a form of democracy. As far as I can tell, when folks say that, they mean "we are a representative republic, not a direct democracy", but "representative republic" and "direct democracy" are both examples of democracies.

So I don't understand where that talking point comes from, why people say it, or what they think they are proving. I note that I see it more often from American conservatives, and my only theory is that it is said by people who think that "Democrat" and "Republican" are just team names.

Is that about right?