Faulty battery, bad alternator, or both (photos)

You are on the right track. That diagram is not drawn exactly correctly. The brown and the yellow are not actually connected to the same contacts in the switch. They are in fact separate "switches" within the switch "housing." They do both energize in "start."

The reason they must be separate is because ............picture in your mind..........

The brown is electrically connected to the coil. With the key back in "run" and the engine running, the coil gets voltage through the ballast, "less than" 12V So the brown "has" this same voltage on that wire

Now if the brown and yellow were actually interconnected, this coil voltage would "back feed" over to the yellow and "attempt" to power up the starter relay!!!!

Now, on a car new enough to have a clutch switch, or an automatic with a neutral safety switch, this could not happen, until you either pressed in the clutch, LOL, or had the car in park / neutral

Even if there's not enough power there to pull in the starter relay, it still will "load down" the coil circuit.