[int_19h]

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.

[wildmusings]

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.

[int_19h]

In that sense, our system is still a democracy. Even the Constitution can be arbitrarily amended with a large enough majority, after all.