Do you want a dictionary definition? Here’s the one from Google’s built-in dictionary.

de·moc·ra·cy

/dəˈmäkrəsē/

a system of government by the whole population or all the eligible members of a state, typically through elected representatives.

You seem to think that you have some technical claim to correctness. You are in no sense correct, technical or otherwise. The reason that people probably get annoyed by your claim is that nobody likes a pedant, but a pedant who is not even correct by some pedantic argument is particularly irritating.

Moreover, nobody ever seems to subscribe to this weird definition where democracy is conflated with direct democracy except people trying to argue that the United States is not a democracy. I suspect motivated reasoning.

Explain how the US isn't a democracy without providing the definition of "direct democracy" and renaming that form of government as democracy to suit your argument.

In theory, I think it's possible for an anarchist republic to exist, where there are no elections or rulers, and also no monarchy or "private" interests owning the state.