People really should try to avoid playing formalistic word games. A republic such as the U.S. is "democratic" in the primary sense of the word because the bulk of the citizens in the country have a direct role in selecting the ongoing governments in the country. That is what makes it "democratic" and is, in general, the fundamental criterion for determining whether a given country is "democratic" or not.
To try and play games with terminology by insisting that "we aren't a democracy, we're a republic" is silly, in general, because on the level of this fundamental question its a distinction without meaning.
To say that there is a difference between direct democracies and representative democracies is true enough, but in either case, the common element is that both are "democratic" in the fundamental sense that the citizens play a meaningful role in determining who exercises the powers of government.
