I has a similar experience with my 79300SD. I noticed that the Battery light did not iluminate when I turned the key to position 2.

My alternator was replaced 1 year ago...turns out that the voltage regulator was faulty. Replaced and all is well!

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'83 240D (W123) 321K miles
250KM Mileage Award
500KM Mileage Award

'84 300TD (W123) 215k miles
250K mileage award pending