The pin 3 / 6 swap doesn't matter for this board. They both tie to ground on the power supply.

What Williams *should* have done is tie solenoid ground to both pins 2 AND 3 using paralleled wires.
And they should have also tied lamp ground to both pins 5 AND 6 using paralleled wires.
This would have helped J2 to be a bit more reliable but J1 would still be a PITA.

Now ... if the one going the the lamp circuit is the one that shorted, check for shorted lamp sockets/wiring.
Maybe too large of fuse?

I also see a 1979 date code on the main filter electrolytic cap in that photo. Time for a refresh for those nearly 40 year old caps.
Rather than replace the power supply with a less reliable supply (imho) - wouldn't take much to get that one back into shape.