Another acceptable term for Representative Democracy is Democratic Republic, You parsed, I lost. I should have listed all of the acceptable alternatives. The power is supposed rest with the voter and be filtered through the elected representatives. At the start we were supposed to elect our representatives who, in turn would elect the senators who, in turn would then select the president. We've evolved a long way and are more of an amalgam of type and certainly not a republic in the strictest definition of the term.
We've slowly, over the years, enfranchised more citizens. At one time only certain peoples were entitled to be elected and there was a tiny enfranchised group to be allowed to vote. Now, we've given the right, not the duty, to vote to a huge number of citizens. We, America, has morphed closer to democracy than republic over the years. The electorate is even given, in some cases, the right cast a, more or less direct, vote for our president. In other cases the members of the Electoral Collage are not bound to follow the voters who selected them.
So, yes our process, these days, is leaning more to democracy than a republic I believe. In any case we are far removed from our roots. Perhaps, as in 1954 (under God added) we will modify the Pledge again. The pledge itself didn't exist until sometimes in the 1890s, my memory is not what it was back then.
I'll stick with my statement, only amending it to include Democratic Republic as an acceptable description of our sometimes messy and always entertaining choice of governance.