Correct on your #568. This is also why the civilian unit is a much superior unit. Some of the modern civilian ones can use an external remote sense, but if nothing is detected on this wire they default to the internal setpoint. The military one would instead just go full field because it thought system voltage was super low.
On #566, you would only be able to do a running re-engagement in park or neutral. Even by not using this wire, you will (should) not be able to crank the starter in any driving gear, only in neutral, due to the neutral safety switch integrated into the shift tower. The starter design itself would not prevent a running re-engagement attempt. The pinion would not engage, but it would try pretty hard to as it is being forced against the moving face of the ring gear teeth.
#566 is typically connected before the diodes inside the alternator, so it is sensing an AC output, since that is what alternators make, prior to the current being rectified into DC through the diode bridges. Therefore, it is easy to see an AC voltage sine wave and judge the engine is running. Not as easy to do with DC. AC alternator connectivity is a thing of the past, but it was used for double-start protection, a frequency signal to run a tachometer, and/or an hour meter. Some modern units still offer this AC terminal, but only got be able to back-service older vehicles.