I've read this many times, and it still comes across as nitpicky and hair-splitty. Democracy is known in comparison to tyrannies--as a common-sense set of ideas rather than some diagnosis proferred by an undergrad in political science 101...
Votes, elections, representation, individual rights--these are associations with democracy. The associations are meaningful enough for a sensible person to say that we live in a democracy.
Shorthand? Maybe. But good enough for government work.
may I suggest reading the following: Democracy VS Republic