Democrat vs. Democratic

I've seen folks say "Democrat Party" and getting told it's "Democratic Party." And seen them respond that they'll start using that name when it starts behaving that way. But the name is a problem. Who to believe?

You have a choice. You can believe the Democrats, or you can believe... the Democrats. Yes, they want it both ways.

[It's taken me longer than I care to admit to find the post linked to, but the content in it was memorable.]