Hmm, I believe we have a misunderstanding.
When you first put your key in, it is in the "OFF" position. Nothing happens here.
One click to the right puts you in the "RUN" position. This enables the fuel injection system, hour meter, and turns on the oil, battery and glow plug lights.
One further click to the right (as far right as the key will go) puts you in the "START" position. This engages the starter to crank the engine, and also turns on the high coolant temperature and low fuel quantity annunciators.
Releasing the key from the START position and allowing it to return to the RUN position (under spring pressure) extinguishes the temperature and fuel indications, unless you have an overheating engine and/or low fuel condition.
Or, at least this is how my 332 works. I know this because I always verify all the idiot lights are working during startup. Perhaps earlier or later years are different?
Just to make sure I wasn't imagining things and potentially giving you bad advice, I went out to my 332 and tried it. This is with the PTO switch ON (to inhibit actually cranking the the engine) and with the key held all the way over to the START position, WITHOUT waiting for the glowplug light to go out. Shown here are low oil pressure, battery discharge, high coolant temp, low fuel quantity and glow plug lights all lit up.