I hear a lot of people in technical (predominantly US) forums make this distinction. Where does the perception that Republic or democracy are conflicting concepts come from?

Republic and democracy are orthogonal concepts. A republic is a form of government where the head of state is elected. So it is in conflict with a monarchy, where the head of state is passed by inheritance. In democracies decisions are made by the people through various means, like elections. In fact many (most?) republics are democracies, where the head of state is elected by the people. The US is one of them, the "the original" one Athens was as well.

However there are republics where the head of state is e.g. elected by a clique of oligarchs or the military. They are undemocratic republics.

A representative democracy is still a democracy.