Thank you. I hope to publish here a short video of the engine running once the new fuel pump is installedAgree with funtimes about the corrosion at the female lead(s). Needs to be attended to before fully troubleshooting the issue.
Based on your information and the wiring diagram provided it appears the ground for both relays is the same and tied into the harness ground circuit, not controlled electronically/anything special. The wiring diagram doesn't exactly match wire color(s) you mention however the system in question is not complex. It has constant 12 volts + and constant ground. The relay is energized by the ecm to make contact between the constant 12 volt + and the + lead for the fuel cell assy.
When you say by adding/jumpering a ground the relay works and provides + voltage to the fuel cell it appears you have found the issue, just need to clean/solidify the ground and repair/replace the faulty fuel pump cell/assy