There is a difference between a "democracy" and a "representative democracy". You can see that I said "representative democracy".
Representative democracy (also indirect democracy, representative republic, or psephocracy) is a type of democracy founded on the principle of elected officials representing a group of people, as opposed to direct democracy.[2] Nearly all modern Western-style democracies are types of representative democracies; for example, the United Kingdom is a crowned republic, Ireland is a parliamentary republic, and the United States is a federal republic.
United States is a federal republic
from your own quote