A pickup is a transducer, that moves energy from one form into another. A conventioal pickup generates electrisity by virtue of a conductive wire moving in a magnetic field. A contact mic works like a telephone.

The sort of pickup you are thinking of and a contact mic both do the same job. They catch the noise that the guitar makes and convert it into little packets of electricity that can be fed down a conductive wire.

These packets of electricity contain "information" (I'm using metaphor now..clever eh) that can be amplified by an amplifier and turned into sound by using the amplified "packets of information" -from, yes the transducer (or pick up or whatever else you want to call it) and turn it into waves of compressed gas phase molecules that are known by lay people as "sound waves" and providing you are not in a complete vacuum and have your **** down, can sound quite pleasant.

SO, armed with the above information, it is obvious that a contact mic is a form of "pickup" as it "picks up" vibrations and helps relay them to a amplifier

QED