To check Spook’s suspicion about the pump relay you can connect a volt meter or test light to the pink pump wire in the trunk. Apparently the pump works OK when you repetitively prime it BEFORE cranking the starter. But it’s hard to hear the pump when the starter is cranking. The volt meter or test light will tell you if you’re getting power to the pump DURING cranking, Lack of voltage to the pump during cranking could be caused by various things - like a bad relay, a weak battery that’s sapped of too much power while cranking, weak impulses from the crank position sensor, to mention just a few.
The one and only time I replaced a Regina pump I failed, on the first attempt, to get the pump properly locked into the sender unit bracket so the line pressure just above the pump pushed it downward, causing a large leak and very difficult starting.