We vote, we count votes, and the person who gets the most votes takes office (with rare exceptions like Gore in 2000 when Gore got more votes in Florida). That's democracy.
That's direct democracy, which we do not have except in limited instances. Your example of the presidency is an excellent example of this, actually. Are you aware of this thing called "the electoral college"? When was the last time you voted for members of the electoral college? Okay, so the POTUS election isn't an actual "one man, one vote" type deal in the direct democratic sense. Plus, it's winner-take-all for each state and thus not even a true representation of how the various electoral college members actually voted. So, not directly democratic either even in the limited arena of the electoral college.
Okay, so how about the supreme court justices? Who did you vote for during the last election? Or hugely influential people in the various cabinets such as Secretary of State. Who did you vote for?
Huh. Okay, so while the US has some parts of government directly democratic ("one man, one vote"), there were deliberately set in place those checks and balances (a constitution and republican structure of other parts of government) to thoughtfully and precisely limit direct democracy, as the Founders felt that direct democracy would be too damaging ("tyranny of the majority" for example) and unwieldy to boot.
The example CauseBy gave, which you say is direct democracy, is not direct democracy. In a direct democracy the electorate vote on policy initiatives. In CauseBy's example, the electorate elect representatives.