There are two synonymous terms for screwdriver, "Schraubenzieher" and "Schraubendreher". "Schraube" means screw. The "driver" part of is translated to "zieher" meaning "puller", or "dreher" meaning "turner".
The word "Schraubenzieher" makes no sense to me, because it can not pull screws. It is even a tool that can not pull anything at all. And screws are things that can not be pulled in a way that is related to being a screw. (A screw can be pulled like a nail, but that means not using it as a screw, and requires a different tool than a screwdriver.) More so, it seems to me that a "screw" is a development based on the simpler device "nail" to make it not possible to pull it. That a screw can not be pulled is not only a property of a screw, not only a prominent and fundamental property, it is the whole base of it's existence.
I see that "Schraubendreher" is the more technical term, and "Schraubenzieher" is somewhat informal. But independent of that, the word "Schraubenzieher" exists.