I've always considered it a "jig" .... if it has a lead/metal head with a hook protruding from it ... regardless of what the hook is covered with. If the hook is bare, I usually refer to it as a "jighead" ... then qualify it by what I put on the hook : hair jig - feather jig - tube jig - jig/minnow - etc.
If it's a plain hook with a plastic body threaded on, and no lead/metal head on the hook .... then, no, I don't consider that a "jig", either. I'm not sure WHAT I'd call that
... cp