[mithuth]

I had no idea there was a clutch lever switch, I'll check into that.

As for the solenoid, I have not opened it to clean inside, only the external points. I will look into removing it and bench cleaning it. That sounds really simple to do.

As for frequency, it seems to be 9 out of 10 starts it just clicks (scenario 1).
Here's a few I tried but the bike had no change:
Bike in N, rocking, kick stand up: no start
Bike in N, kick stand down: no start
Bike in gear, rocking, clutch pulled: no start
Bike in gear, clutch out, rocking, pull clutch, push start button: occasional start
Bike in gear, clutch out, rocking, push start button, pull clutch: always starts

I'm leaning heavily towards it being that solenoid, so I'll try cleaning it and putting it back together.
If that don't fix it, I'll try replacing it.

Where is the clutch lever switch located, in the rubber boot up on the handlebar. or down on the ending somewhere?

A thought I just had, if the battery is weak/weakened or not fully charged, could it be sending not enough current to the starter until I rock the bike (alternator boosting the charge)?