Hmm.
So let's outline terms to make sure we are on same page.
Male Pins 13 and 14 are on the relay board. Female sockets 13 & 14 are on the wiring harness.
Socket 14 on harness brings power into the relay board direct from battery and transfers it INTO the relay board via male pin 14 on the relay board.
Pin 13 is 12v going OUT of the relay board once the relay is energized. This is transferred to pin 13 socket of the wiring harness to the fuel pump.
if you used the Test light light to probe the back side of the harness with it plugged to the relay board and had 12v there, the pump SHOULD have been running.
I don't really like test lights:
Here's why. if you probed the harness with the test light and it lit, you would ASSUME that you have 12 volts present. However, you would be wrong in assuming that. You could have 10v and still light the test light.
Why does it matter?:
You could have had the relay energized but inside the relay you have a high resistance contact between relay pins 30 and 87 due to arcing, wear, or corrosion. The relay could be closed contacts just enough to have a voltage present but not closed well enough to actually carry sufficient fuel pump motor current to run the pump.
A DMM would be more likely to pick this up by showing you something like 9V or 10V indicating that the high resistance voltage drop is there. A test lamp won't necessarily pick this up.
So what:
it very well still could be that replacing the relay is the solution. However, it could also be that the relay socket contacts are the place where the high resistance connection was occurring (not inside the relay) and the act of using the jumper allowed a good solid connection to occur temporarily and when you go back to the relay, you'll have the same intermittent lack of fuel pump operation. You will then go on a goose chase assuming that the relay was replaced and all is good when in fact, the problem still lies with dirty contact in the relay board relay sockets.
Likewise, there is a possibility that the problem is in the relay board traces and the simple act of disturbing things (replace relay with jumper) is enough to get it working temporarily.
This is the hard part of electrical testing that requires patience and a logical method to get to the root cause of the problem. What's next:
If it were me, I'd try to see if you can duplicate the condition again where you have power on pin / socket 13 but the fuel pump not running and test some more using the DMM.
IF I could duplicate the condition, i would disconnect the connector at the fuel pump. Would verify I still have 12v at the pump. if i did, I'd switch my DMM over to 10 amp current measurement and probe between the 12v source (wiring harness) and the 12v terminal on the fuel pump and verify how much (if any) current is drawn.
There could also be some sort of funky ground side connection going on but let's not go there yet.
@second wind