Check your message inbox for a machine electrical schematic.
Cat used systems manufactured by this same "Prelub" outfit on larger tractors and on off-highway trucks. In some cases (large tractors) the prelube pump was actually part of the starter motor. In others (trucks) it was a separate electrically-driven pump like yours mounted on the RH frame rail ahead of the front suspension mounting.
If this system works like the ones I have seen before then it either runs until a certain oil pressure has been reached, or until a certain amount of time has passed. In the case of the latter a timer cuts the power to the prelube pump and sends it to the starter just like it would if the specified oil pressure had been attained. If your system has this timer installed then that could be what's not functioning correctly, or you just may have a bad electrical connection somewhere. See attached.
So studied that schematic some and it only feeds enough oil to lube the main oil gallery, mains , rods and maybe the cammshaft bearings as the cut out switch is cutting off the ground at 4-7 psi which is more than enough for rollover protection prelube.
On his system I see the picture of the relay so bet the installer opted for an oil pressure switch as it is way cheaper than a cat timer with all the wiring aaasociated with the timer.
Most Forget to Remember that those kinds of pumps can only be used with oil pressure switches that are set very low unless you use 0W40 oil only. Why, because even in Texas if you were using a thick oil on a 30-40 Degree F winter morning the electric motor will draw way more amps than the Small solenoid can handle.
They burn out very fast from the arc when the solenoid contacts release.
If he follows the hose feeding the Oil gallery most likely, he will find a Tee with the pressure switch on the tee being fed the oil from the pump.
Would be good to see another picture.
Simon C