Yes this seems to be a common distinction made in the US, which I also don't understand. What I learned in politics at school (and studying it for a short time) was that republic and democracy are orthogonal concepts (leds leave out the federal which seems to be even another dimension).

A republic essentially means, the state doesn't have a king (head of state by inheritancel, but some sort of president which gets elected in some way (not necessary by the population). A democracy is a category of how decisions get made, i.e. by some vote of the people (demos).

Is there some subtlety I'm missing or is this thing about "federal Republic not democracy" something just always repeated, without properly understanding it. .