What Draugnar describes as true democracy is direct democracy, which is not actually a true democracy because there is no true democracy. Democracy is simply a form of government where the power legitimacy is ultimately traced to the will of citizens, as opposed to the will of the ruler or God or whatever. Direct democracy is just one of the ways to accomplish that; in fact, it was a supplementary tool to representative democracy in Ancient Athens already.
On the other hand, the precise nature of a republic has been in question for 2,5k years. Colloquially, a republic is a representative democracy with elected leadership. Which of course doesn't cover all the bases since that would apply to tribal military democracies, a very common form of government throughout history, which is nevertheless never classified as a republic but rather as an elective monarchy. It would also apply to Roman dictatorship as well as Cromwell's Commonwealth, governments that go against the notion "republicans" try to pass over (they are autocratic). On the other hand, many regimes labeled as republics also goes against that notion, like merchant republics of Italy (plutocracies), noble republic like Polish-Lithuanian Commonwealth (oligarchy), Islamic republics (where the law system is preordained by God and not people), Socialist republics (which lack political pluralism) or republican oddballs like Syria or North Korea (monarchies).
The point is the USA is a representative democracy AND presidential republic at the same time. Saying it's only a presidential republic doesn't differentiate it from Syria, saying it's only a representative democracy doesn't differentiate it from the UK.
Also democracy isn't intrinsically a rule of a majority. That is a majoritarian democracy; again that's a parameter of democratic government, not a feature of democracy itself.
Ultimately, what is a true democracy depends on one's attitude towards the process orientation vs result orientation. Process orientation might view direct consensual democracy as the true democracy, while result orientation doesn't really care about the form of the meatgrinder through which citizen's voice transform into action as long as it accurately reflects his opinions. I preffer the second, no non-sense approach hailed across political spectrum by "reds" as defined by Putin in a certain thread as well as the "silent majority" of GOP. How much do they succeed is of course a different point, but the ideal is there, unlike in tunnelvisioned minds of anarchists or tea partyists.