@Chaqa, in common parlance, a democracy means either a direct democracy as you describe or the modern representative democracy which we have in the west. This as defined not by your idiotic american schools, but by the common usage of the people.
Since we have very few direct democracies in the world today, we usually talk about representative democracies. So even though it is not specific, the term usually refers to the kind of voting and representation that we have in Ireland, or the UK. And that you have in the US...
It only becomes hazy when you look at the likes of China and Iran, which have elections for their rulers, but the party and/or theocratic council have to approve all the candidates. Thus you aren't necessarily represented by a person who you feel represents you, only by someone approved to represent you. (though i think China has more in common with the US than the US has with the UK or Ireland, given the lack of voter choice in the US so clearly highlighted by this year's election... Iran is further towards Theocratic rule, as despite having an elected President, the President is not in charge, and the Supreme leader is not elected, except once when the last one dies... )