The problem is as soon as you limit the size of what you consider the people deserving the power, you limit democracy.
Your view seems to agree that if only citizen are considered as the people allowed to have the power then it is fine.
But then, if someone consider the king to be the only people allowed to have the power then it should be fine too.
That's just a different scale after all.
So if one is a democracy to you, then the other one, were only one person has the power, is a democracy too.
Which would be quite paradoxal don't you think?
I don't classify X or Y as democracy. I just use the definition of democracy and see if it fits X or Y system.
What we have now is a representative system (which is the opposite of a democracy) elected by some.
You and many, including myself occasionally, will call it a representative democracy, but it is a semantic slippery, as the very creators of our current systems, such as Madison in the US, and their relatives in UK or France (which were the 3 first countries to use an elected representative system) intended.
All of these founding fathers were afraid of the masses and what choice they could make. And so all the respresentative system was designed to limit the democracy, yet remaining inclusive enough to be still be able to become the norm of what we call democracy.
You won't find any constitutionalist that wouldn't agree wiith that : we do not live in democracies.
(And probably never ever will, unless we reach some sort of global hive mind)