"Why do I immediately sweat more when I drink cold water after a hard run?"

Was it because of the change in internal temperature?
Or because I had more fuel for sweat (i.e., water)?

Looks like the #1 reason is the "feed-forward signal" that removes the sweat suppressing brake.

Your explanation about the "feed-forward signal" makes sense. The body anticipates the need to maintain thermoregulation, triggering sweat production in preparation for the temperature drop from cold water. Smart biology at work!