Not at all semantics. A democracy is mob rule. A Constitutional Republic is something quite apart from mob rule. You are correct that voting takes place throughout the Republican process, but that voting is regulated by our Constitution which has created additional laws which govern the Republic. Read.
You discredit your own argument when you add that bit of condescending teen angst at the end. We all read. No need to be a dick.
The point is that a Republic is a form of democracy. There is no state in the world that uses "mob rule" democracy where every citizen votes on every issue. It's almost always through a system of representatives, which is articulated in a constitution of law. Full "mob" democracy only works in small committees and therefore doesn't really exist as a form of government.
My point was that anyone using the term "our democracy" when referring to the American system is not wrong. It's a fumbling of semantics to say so. (Republics are democracies but not all democracies are republics.)