After we bought the '96 E300 I downloaded a .pdf owners manual. It says this:

"If the preglow indicator lamp fails to light up, or lights up while driving, the preglow system is defective . . . "

When we got the car the CEL/MIL was lit. The preglow indicator behaved normally, IIRC. It woud beep and the multifunction display would read "check engine electronics" upon startup and again at shutdown. The OBD tool brought up code P0380. #1 glow plug was bad. I replaced all the glow plugs and reset the ODB system and everything has been normal since.

A wiring fault will also trigger P0380. You might test the "bad" glow plug at the head to eliminate the possibility of a wiring issue. Which plug is it? Maybe it's one you can reach without having to pull the intake.