Your terminology is not consistent with my industry (oil refinery). Any pump can be driven by any driver. We have centrifugal and positive displacement pumps that are driven by air motors, electric motors, steam turbines, gas engines and diesel engines. We have pumps that are mechanically driven from other equipment by being coupled up to a gearbox, pump, compressor, motor or engine. We don't have any pumps that are driven by hydraulic motors, but we are considering hydraulic motor drivers for some of our decoking equipment. I would not describe any of our pumps as electric pumps or hydraulic pumps. I would describe them as a centrifugal pump, gear pump, screw pump, piston pump, diaphragm pump or progressive cavity pump.
If I mentioned a hydraulic pump, I would not be speaking about the driver or even the pump configuration. I would be describing the service. We have pumps that provide hydraulic pressure for pistons and actuators. These pumps might be called hydraulic pumps even though most of them are variable displacement piston pumps driven by electric motors. Some of them are gear pumps driven by electric motors or steam turbines.
If I ever referred to an electric pump it would just be because I was too lazy to say electric motor driven pump.
Other industries might use different terminology than what I am describing.
Johnny Pellin