Yeah, I wait for the glow plug light to go off, in the morning(if it's the case). If I know the engine is warm, I start the engine in an instant, without even looking at the light.
My car does what I describe above even with a new battery, so that means the starter is at fault. After viewing a video of my engine cranking, adams24 said he never saw a v6 tdi crank that slow before. I haven't paid attention as I didn't used the car in a while.