Hey, RG1, I'm hoping you're still monitoring. Here's what I have found with that method of testing you mentioned (and more new stuff today).
I got my mower ready for this season and, of course, the battery was dead. I jumped it off as usual from my vehicle (was in a hurry) and got the one-click turn that eventually started the mower (ran like a champ after). After mowing, it did the same delayed shut off after turning off key (about 5 seconds of idle, then shut off).
I remembered your thread the next time and disconnected the battery, connected jumpers to corresponding cables directly and tried to start it up. This was weird. The starter gave me a "rat-tat-tat" click sound. Three consecutive clicks, but no turn. I didn't want to try that so many times, so I connected the battery again and jump with the same results as the first part of this post.
That brings me to today . . . I started with the usual method, but stopped mowing near the house to clean the deck out (grass, etc). It wouldn't start back at all with a jump.
I do still get the 3-4 error code (electrical) and checked all connections to the starter, ignition, etc. and all looked in order. Double checked fuse too - all good. I would get occasional jolts when turning key, but no turn over from starter.
So, there you go. I'm looking forward to your post (really digging this forum and telling everyone I know!). Thanks for any insight you can give, as always.
3-4 error code via JD Manual:
Starter Output Short to Battery
This code indicates that there is a short in the starter circuit to the battery.
• Inspect electrical harness for pinched, chafed, or frayed wires.