I think you're getting lost in semantic games. Cloud providers sell you access to their computational resources. When you pay for a server, you're paying for the right to access that server and do what you wish to do with it. Much like when you rent a car, you still pay for it if you keep it parked.
This is true, but it still may be how Hetzner decides usage. I'm not sure if this is the case, but it seems pretty reasonable when you're talking about bare metal.
> Do you bill servers that are off?
> Yes. Until you, the customer, delete your servers, we will bill you for them, regardless of their state.