Openly admitting to being pedantic. That said, here we go: it's a DRIVER, the end is the CHUCK, which holds the SHANK of the BIT, which can be a DRILL.
I understand it's a setup to the punchline, and nobody actually gives a shit.
A drill and a driver are separate tools. You can fit driver bits in a drill's chuck, but it's not its intended purpose. A driver is meant specifically to drive screws, and it's better at the job than a drill.
While this is only semantics and based on nothing more than my preference, I would argue that a drill is a type of driver. If you call it a drill driver, I might get upset though.