Republic refers to any form of government which is not a monarchy.

The US is a republic -- specifically, a federal republic -- it is also a democracy -- specifically a representative democracy.

The whole "The US is not a democracy, its a republic" thing makes as much as saying "16 isn't a perfect square, its an even number."

Neither does saying "republic". And "representative democracy" isn't succinct compared to "democracy", its verbose.

> Saying democracy instead of republic just undermines the common understanding of how our government was setup to function.

No more than saying "republic" instead of "democracy". A "federal republic that is also a representative democracy governed under a written constitution" would avoid that problem, but its a lot to write, when "democracy" is the part that is relevant to the point you are trying to make.

A democratically elected representative republic, specifically. I agree that you are correct in a pedantic sense, but our society is a "democratic society" in a more general, looser sense. Given the context of TFA, I think the usage is correct enough.

The United States of America has a government which is both a republic (and, more specifically, a federal republic) and a representative democracy.

There are republics that aren't any kind of democracy, and representative democracies that are not republics (e.g., the United Kingdom.)