No, the distinction is that the Athenian democracy was a direct democracy where every citizen could vote on legislation, as opposed to representative democracy

No, the distinction is that the Athenian democracy was a direct democracy where every citizen could vote on legislation, as opposed to representative democracy where representatives vote in a legislature. The US was a democracy, albeit imperfect, even when slaves and women couldnt vote. There are still some remnants of direct democracy: ballot measures and New England town meetings.