I ran into this just the other day when somebody commented of mine in which I talk about 'liberal democracy,' meaning the philosophical term not a description of a practical kind of government setup. He told me the United States isn't a democracy but a constitutional republic.
I responded that the Federal Republic of Germany, the United Kingdom, and the United States are all liberal democracies, even though each of them has a different constitutional arrangement. Just because the UK is a constitutional monarchy doesn't mean that it isn't at the same time a liberal democracy. Ditto for the United States.
I didn't really understand his comment then, but I get it now because I'm seeing it more. I'm remembering that when I was a kid in the 1970s in a very conservative Baptist household, I'd often hear the same thing about the United States being a republic and not a democracy, and I think for much the same reason.
Thanks for an excellent article explaining it all.