Battery Pack

[Dan schultz]

That really needs cleaned up, and make sure the ground side is still connected. The batteries are used for alternator restart in the event they get knocked offline. The restart procedure is in your POH and spelled out in the maintenance manual.

Your alternator out light is more of a function of the board assembly that is on the front of the firewall by your regulators. On this board are two big diodes behind it and on the front are smaller diodes, resistors, and two rectangular reed relays. This whole board assembly is what turns off and on your alternator out lights.
Much info is available on other threads.