I have the electrical diagram for the '82 300D. (The '83 could be different.)
It shows that the coolant temp gauge is powered through terminal #6 (blue/red wire) on the instrument cluster round plug. It is grounded through terminal #2 (green wire, running to a connector on the left front fender well, then to the coolant temp sender.)

Correction: The coolant temp gauge is powered through terminal #6 (black/blue wire.)