The only wiring schematic I could find for your car is for the 2000 Buick Regal, but I hope this helps,
since it gave me a reason why starting a car would cause the driver's power window to
go down, and no other window in the car - it has a special circuit component
(the "Express Down Module").
Its possible that the Express Down Module in the power window switch circuit has failed. This
module latches the "Express Down" switch in the "made" state until released by the "Express
Down Module." If the Express Down Module is latching the Express Down switch when it
shouldn't (at power-up) instead of when it should (at the window "down" limit), then the driver's
window would lower when the driver starts the car. The way to tell if this is the problem is to
note how fast the driver's windows goes down - if it goes down ALL THE WAY, then you know
that the "Express Down" circuit is involved (and therefore suspect the Express Down Module).
If the driver's window only lowers part of the way, then the Express Down circuit is exonerated.
See the various components of the driver's power window circuitry in the
2000 Buick Regal power window circuit schematic below.