We are a representative democracy and a republic. Those things aren't mutually exclusive.
When people say things like "US is a republic, not a democracy", they're using the archaic definition of "democracy" from 200 years ago. Modern English has a different definition.
I think that "archaic" definition still has plenty of modern salience. Whenever someone is complaining that a majority of the people support xyz or wanted so-and-so as President but didn't get it, they are appealing to "democracy". Whenever someone responds by saying that's just fine and that the point of the system isn't to translate majority wills into policy, they are appealing to a notion of republicanism.
The modern terms for these is "direct democracy" and "representative democracy". Furthermore, insofar as the ability to decide things by voting is constrained by the constitution, it's a "constitutional democracy".
There are many countries in the world that fulfill all these criteria without being republics. Canada would be the closest example - they have a constitution that limits democratic decision making, and they have people elect representatives rather than deciding on issues directly. But they're not a republic, solely because they have a monarch.
Those are two totally different meanings of the word “republic”. The definition as the opposite of monarchy and the definition having to do with organizational qualities are related only by historical overlap, not by unity of core concept.
And what I’m trying to get at as one definition of democracy is the idea that policy should track majority preferences, totally aside from any structural features of the system that might make that likely. For example, someone arguing that a certain bill should pass because a majority of citizens support it is making an argument from this idea. That argument in itself is totally aside from whether the legislature is even elected by the citizens.
The definition of republic has held up. It's still true that we are a republic, even under modern English. Some call it a democratic republic but notice no one calls it a republican democracy.
But it's usually redundant, because all modern democracies are representative. Some have more direct democracy features than others (e.g. Switzerland, or some western US states with their public initiative process), but they're still fundamentally representative.
We don't call this a representative democracy. We're just not a democracy. Majority vote doesn't even determine the president. Even in the house and the Senate, bills have to pass BOTH in order to become law...and remember, Senators don't line up with population figures...they represent whole states. The definition of republic is 2000 years old. It's okay to use it.
The US is (by Constitutional design) both a representative democracy and democratic, federal republic.
“Republic” alone is correct, but overly nonspecific; essentially any government that isn't a hereditary monarchy or similar system where government power isn't private and heritable is a republic.
> The definition of republic is 2000 years old.
Well, sure, there is a definition that old (older, actually), but the one you seem to be using (the colloquial American one referring specifically to a system of elected representatives) is much newer, peculiar to American non-technical usage, and equivalent to “democratic republic” and very close to “representative democracy” (differing only in that the latter can coexist with constitutional monarchy.)
We have many Democratic elements. We elect our Representatives, many times we elect our own laws, and we even indirectly elect businesses, voting with the wallet. But yes, generally speaking we live in a republic hybrid, for good reason.
Electing representatives to manage the government is the defining part of a republic. When citizens are voting on laws themselves, they are practicing actual democracy. That's relatively rare in the U.S. and it also only happens at the state and local level.
When people say things like "US is a republic, not a democracy", they're using the archaic definition of "democracy" from 200 years ago. Modern English has a different definition.