The Roman and Greek republics were democracies but the wealthy controlled everything and legalized things like seizing individual lands and slavery.
We seem to be operating with different definitions of "democracy." For my money, you're not a democracy unless every adult gets a vote. So, for instance, the United States was not a democracy until the Voting Rights Act was passed in 1965; prior to that, either women or minorities or both (at varying points, it would be one or the other) were disenfranchised.
So, no, I don't think it's fair to call ancient Athens a democracy, even if that's what it called itself. It wasn't rule by the people; it was rule by a certain segment of the male population.
A liberal democracy is not characterized merely by elections.
It must have two features
-first, the law and its enforcement represents the will of the majority of the population (hence, the elections)
-second, this law applies to everyone; no one is above it, not even the current rulers.
There's a relatively recent term - anocracies (or crappy governments).
It includes states that call themselves democratic, but really aren't. Why?
Because, in such states, the ones that win an election are no longer bound by the law; they can pretty much do whatever they want, much like the princelings and kings of history.