Did the water pump have play in the shaft that you noticed AFTER you pulled out the old pump? I cannot notice much play at all in the pulley.
I took it to the dealer and they replaced the tandem pump (self-leveling/power steering pump) and belt tensioner all under warranty but it didn't change a thing.
The A/C compressor, alternator, and smog pump are only 6 months old. Also, why would the noise be more subdued once the engine is fully warmed up? Makes me think more and more that it IS the water pump. BTW, this is the original water pump from this engine (1992 M119.960 engine).