If you're in a hurry just take a thick black wire and connect it to the solid black wire and the other end to a good spot on the frame. Ground is ground.

 

 

A couple of things.

1) There are two sets of two wires going to the pump assembly:
Power for the pump motor from the relay fuse box = Gray
Ground for the motor = black out to a splice pack on the frame and then black again to the frame ground

Your best bet here is check battery positive to battery negative during pump on and then compare that to probing gray and black right at the pump during pump on. It that net voltage to the pump is too low then we figure out if its the battery feed on the gray or the quality of the ground. If you want to check the quality of the ground now that ohm reading from pump motor black to a clean spot on the frame.

2) There are two wires for the fuel gauge sending unit:
Purple = the send signal for the computer
Black out to a splice pack that shares a connection with a black/white stripe into that splice block but that is not important right now.

If you really have a net 5V from gray to pump black right at the pump at pump on then we need to figure out why before you install a new pump

George

 

found here: 2000 S10 Fuel Pump Wiring - wikiduh.com
I believe it is worth checking to be sure it's not gonna fry Your pump.
Personally I don't thing Your pump could develop 60PSI on 5V power supply. It would have worked with less than a half of nominal power. And not holding the pressure doesn't sound like an electrical malfunction either. Is here a chance You've been measuring accidentally the fuel pump switch/sender maybe?

 

Gosh, I woke up today with an awareness I've made a terrible engineering mistake;
Power[W]=Voltage[V] x Current [I]
It is not only the voltage, but also the current that will vary. A constant is the load value.
So, let's assume the pump has a 20W. From the above equation we may calculate the current that flows through to find out the load:
I=20W/12V=1.6A

load resistance would be
R=Voltage/Current=12V/1.6A=7.2Ohm

Since we have the resistance, we may check out what happens if the voltage drops down to 5V:
the current is going to be:
I=Voltage/Resistance=5V/7.2Ohm=0.694A

hence the power is going to be:
P=Voltage x Current =5V x 0.694A=3.47W

The pump power factor is going to be 3.47W to 20W which is 17% of the nominal power.

 

 

Not necessarily. Initially the pump primes in (You may hear a buzz for 3secs once the key turns on). Then, once primed it is unnecessary to run it. Unless the engine rotate (=is being cranked or run).

 

So I will need a helper for that task, got it.

 

The purple 5 volt wire is the signal for the fuel gauge sender. The new pump will come with wiring as well. I'd solder all new and verify power and ground before pluging in my fancy new assembly. Happy wife happy life.