Faulty battery, bad alternator, or both (photos)

Hopefully someone can answer this question.

I figured out what the wires not showing continuity are.

J3 12gauge Brown and M16 20gauge Black.



J3 wire (brown) feeds into the seatbelt interlock connecter and also splits off going to ignition ballast resistor and ignition coil positive (shown as J3 14 BR in the diagram):




M16 (20gauge Black) feeds into the seat belt & key in buzzer (which has been disconnected since I got the car), so that makes sense that there isn't continuity.



What about that brown coil/ballast wire though?

With the key out of the ignition and car in park, shouldn't I be getting continuity in the brown coil positive / ballast wire?