That is an accurate description, but I'll add that you can start the player again after the last step you describe, doing nothing else in between, and it will boot to playing mode. Therefore, the player doesn't think it's in standby mode (thanks - I couldn't think of the correct term) in your last step, as, if it did, it would continue to boot into standby mode.

It continues playing from the correct point; it's not "rolling back". Well, it does every once in a while, but not regularly and certainly not in correspondence with this behavior.

While I do not have a diode in line with the ignition sense, I have none of the other symptoms associated with needing one. That is, I have no amplifier thump, no consistent "rollback", and it goes into standby mode properly when I turn off the ignition.

I'm inclined to say that I need a more powerful battery, but I'd rather get some confirmation that it is a voltage problem before spending $150 on a super-duper starter battery to replace one that's otherwise just fine.