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Actual political scientists use the term to describe the US. Everyone knows what it means as a blanket, general term for a class of governmental systems. Unless there's a good reason to split that hair, they don't, and you shouldn't either. No-one's confused or gets the wrong idea when someone describes the US as a democracy.

I don’t think that distinction is helpful in this context.

A republic is a form of democracy, like a cat is an animal.

> A republic is a form of democracy

No, it's not.

But a democratic republic or representative democracy with republican form (as opposed to, e.g., a constitutional monarchy) is both a form of democracy and a form of republic.

There are autocratic and/or theocratic republics, among others, that are not forms of democracy.

A rebulic is a form of democracy the same way a moon is a form of planet. Instead of orbiting around the sun, it's orbiting around a planet and that planet is orbiting around the sun. Or, instead of voting directly for things, people vote for other people, that vote directly for things.