If you have slaves, then by definition the lowest group doesn't have any right, and what you really have is an aristocracy - though one with an unusually large ruling group. There may be democratic elements, including elections, inside the ruling group, but even a free election doesn't mean the majority rules if only a minority has the right to vote.
This was a distinction the Greeks themselves made at the time—and, what’s more, it’s actually kind of relevant to Douglas MacArthur taking over during a Second Civil War.
In 411 BCE, a group of Athenians organized a coup and took over, saying that the Democratic government was losing its wars through incompetence. (Which was true.) They said that they would instead let only 400 citizens vote; later, they promised to expand this to 5,000. They called this an oligarchy, and at the end of the Peloponnesian War,
democracy meant the system in place at the start of the war. From a modern perspective, we might say that most of the population didn’t get to vote either way, and indeed this had occurred to them: Aristophanes’ comedy
Assembly of Women imagines what might happen if women all voted and took over the government. But, at least to the upper-class Athenian men whose writings have survived, that wasn’t the distinction between
democracy,
oligarchy and
tyranny.
Athenian attitudes toward slavery were complicated. One of Solon’s reforms of the constitution, which by the fourth century BCE were remembered as incredibly wise and just, was to free all Athenians who had been enslaved, outlaw any more families selling themselves into slavery as a condition of a loan, and repurchase and free all Athenian slaves living elsewhere. However, this only applied to slaves of Athenian descent. They were fine with slaves from elsewhere.