It is charging.
From looking at the wiring diagram, the white wire is the ground side of the indicator light.
When the key is turned on with the engine off, the charge light will receive + battery voltage from the key switch. The bulb will illuminate by having a ground path thru the alternator's internal voltage regulator. As the engine starts and the alternator starts producing power, terminal L should start having a + battery like voltage and in this way the indicator bulb now has + power on both of its terminals and will go out.
I have attached two pages from the WSM dealing with the removal and testing of the internal regulator and rectifier.
Before starting any further work now that you have shown that the battery is being charged, can you provide more history on the origins of the light problem. Perhaps you already did but I cannot go back to the first page of the thread without loosing what I have typed.
My inclination is to replace the regulator but you can also do the simple test to see what it reveals. Make certain you do the test with a simple, non auto ranging meter.
Dave