On the other hand, everything is a de facto democracy because revolution is always on the table. There's nothing wrong with being precise. In fact, it leads to fewer misunderstandings.
The problem is that preciseness seems to be something that all US people know (is it taught in high school?) and nobody else. Systems where the government is voted for by the people are democratic regardless of the details.
Similarly when electing the president people vote for a person who will vote for the president. While it is a weird system, in essence the leader is chosen by the people.
Calling only direct democracy a democracy would be weird, by that standard a country pretty much can't be democratic for purely practical reasons (though Switzerland comes close)