Is the light intended to stay on as long as the glow plugs are operating, or is the light going out an indication of 'ok to start'? IDK.
Go see a guru.
Bill
I second Bill's recommendation to go see a TDIclub GURU in your area.
I'm not a guru but I'm somewhat familiar with the GP behavior. On 2002 and 2003 TDIs (ALH engine), the GP light is an OK to Start light. The GPs actually stay on up to about 10 seconds longer if you wait longer. There is no need to cycle the key to get extra glow time.
The 2002s and 2003s have a 4-wire GP harness and a solid state GP relay. You won't hear this relay click because there are no moving part(s) inside and the relay is located under the windshield wiper cowling. IIRC, the change from the older 2-wire GP harness to a 4-wire harness was due to changes in OBD-II requirements which now require individual monitoring of system components (can't monitor components as a group).
Older TDIs prior to 2002 (not sure about 2001) had a 2-wire harness and a mechanical GP relay. IIRC, the GP light on the dash showed when the GPs were actually ON and you could hear the GP relay click on or off.
In my 2002 Golf TDI (sold it in 2010), one trick I would do in the bitter cold weather was keep the driver's door open when I turned the key. Keeping the door open was only to keep the interior lights on while starting the car. When I first turned the key to ON for GP operation, I could see the interior lights dim slightly while the GPs were ON. The GP light on the dash went out but the GPs remain ON for up to about 10 seconds after the GP light goes out. I would hit the starter the instant I see the interior lights brighten back up when the GPs finally turned OFF. This helped starting in below zero F conditions because the GPs stayed hot long enough and lightened the load on the battery during cranking.
I thought about doing a mod to wire up a light to the GPs so that I have a "real" GP light to know for sure when the GPs are actually on or not. I never bothered with the mod because I stayed on top of maintenance and never had any cold start issues with the car during my 8 years and only 361k miles of ownership.