Am I right in thinking the Lt. Blue wire is the hot wire for the washer pump?
The three wires hooked to the wiper button are all used for grounding. The wiper and washer pump are powered off of an ignition source so when the key is on the power is on. To use the switch with your pump it would be a simple matter of running a hot wire off of the fuse panel. You can plug it into the “Unfused Ignition” section and set up an inline fuse if you want to fuse it. Then you could extend the wire for the washer pump switch to the new pump so that when you pushed the button the pump would work. There are two wires on your original pump, 1 is power, (yellow, I believe ) and the other is the wire that you want to extend to your new pump.