Hey now, if you are going to be pedantic make sure you get your facts straight.
If I'm remembering right, prior to James Madison's writings, there wasn't a real distinction between democracy and republic, besides the fact that democracy was Greek and republic was Latin.
Madison in the Federalist Papers made the distinction between democracy and republic that are your talking about, but it is pretty arbitrary.
Now you are right that we don't live in a direct democracy (though states have features), but we are still a democracy and a republic.
Now that I am being pedantic someone will probably point out where I am in error, but I just felt like posting some information for whoever may find it useful. :)