Actually, we really are just a democracy. Where and when the unmodified term "democracy" got morphed into being interpreted as "direct referendum on practically everything" is unknown to me, but it never was a definition that any state in history actually ever met, even the ancient greeks who it supposedly came from. A democracy is just a broad category meaning a state that conducts free and fair elections for public office and guarantees certain rights to association, speech, etc in its social contract. Actually, in Political science research you normally get the label "democracy" just for having free elections. "Republic" is almost the same word except that the people who 'represent' the governed don't have to be (but usually are) directly chosen in an election.