that's the oil pressure sensor, has nothing to do with starting if it's cranking and has 12V to the injection pump solenoid.
kinda sounds like maybe your "jump start" setting is doing a lot of work and tripping after a second or so. And/or the battery is weak (could be killed from cranking too much, could be half drained). And/or you have some bad connections in the battery cables. Or the hydraulics locking up, try the steering while cranking to see if that changes.
I'd swap a known good battery into this, or try a good pair of jumper cables with the ground to the starter body/bolt, and positive to the positive cable terminal on the solenoid. What gets hot after cranking for ten seconds? feel all the connections and the whole length of the cables and the starter and solenoid. carefully till you know they won't burn you, or mist with water to see what "steams".