DIrect democracy = direct rule by the people.
Indirect democracy = the rule of the people is delegated.
Republic = Rule by the representatives of the people.
Constitutional Republic = A republic with a constitution.
The United States: An indirect democracy and Constitutional Republic. The people vote for their reprsentatives, who are then supposed to represent the people. This is indirect democracy and republicanism in action. It's not an either/or, it's both/and. It's true that not all democracies are republics (e.g. the United Kingdom is a Constitutional Monarchy with democracy); and not all republics are democratic (e.g. The People's Republic of China). Technically, a monarchy can be democratic; and a republic can be undemocratic. A republic can be a dictatorship, and a democracy can have a monarchy. The United States, however, is both a democracy and a republic.
This democracy vs republic language is just political bloviation. The goal here is to politicize semantics, and is really intended to be a kind name game: Democrats/democracy; Republicans/republic.
See, Real America™ is a republic, not a democracy, so Republicans represent Real America™.
We love the Constitution.
They hate the Constitution.
We love America.
They hate America.
We love freedom.
They hate freedom.
That's the rhetoric. Don't believe it, it's dumb.
-CryptoLutheran