You're not thick or alone in being confused with this dilema, where you have the necessary voltage but a good working unit doesn't work.
The reason is there is a majorly bad connection, so whilst it can easily read 12.6 or whatever, it can't supply this under load, or a electrical item, the amount a multi meter uses is nothing compared with the pump.
It is more likely to be a bad earth, so you could pull off the connectors and try a substitute live and earth in turn to see which is missing, my money says earth is dodgy.
Missing lugs will NOT stop it from priming first thing in the day, but the prime is a
temp/time related thing so won't do it all the time anyway.
A missing lug will prevent it from running properly, as it will lose the spark for that cylinder.
I would get a bit of thin wire, bare the end about 1" of the wire, and tuck this into the connector plug, and push onto the pump, then connect the other end to earth or live as appropriate, make sure it is a decent earth though, not a cruddy piece of rusty metal!!
tabetha