You don’t get it.

Your definition of “democracy” while in some regards is technically correct it isn’t colloquially nor actually correct. It’s what I would call “40 year old civics class correct”

Because see the definition of democracy does include representational forms of democracy.

“government by the people; a form of government in which the supreme power is vested in the people and exercised directly by them or by their elected agents under a free electoral system.”

What you are referring to is now called “pure democracy”

So see the reason people believe it is a democracy is because it is. No one is pretending that it’s a pure democracy.