The pump turns if 12v is applied to it directly?
Power is reaching the brown wire at the connector for the fuel pump fo 2 seconds ?
I've just copied this diagram from my article here:
Link. The contents of the pdf explains each of the 3 separate circuits.
The fact that the motor turns when it is powered directly suggests that it should also turn when powered by the 'system'.
The power on the brown lead turning off after 2 seconds suggests that the 'switch' in the ECU is working correctly.
(I doubt that it is a switch / relay, but an electronic device that acts like a switch in this case.)
Those two things are confusing. It should work, given that information. So I need to ask some questions to be sure that you are actually seeing what you describe.
Where did you get the power from to make the pump work ?
Where did you apply the +ve and -ve leads ? ie where did you apply the +12v and where did you apply the negative battery feed.
Can you confirm that you tested the 12v at the
brown lead and not at the brown / black lead . You might need to go back and do this again because you cannot always see the black stripe on the brown/black lead.
I am suspecting a dodgy earth lead or a bad battery at the moment, but I need more precise information about your tests to be able to help.