The most effective way to resolve the issue of contested usage in these sentences is to clarify the subject of each sentence without causing confusion. The first and second sentences successfully distinguish between the pitcher (the container) and the players.
The best option is:
"We brought some water for the baseball players. But one of the players said that he wanted lemonade instead."
This version avoids confusion between the pitcher (the player) and the pitcher (the container) by using "some water" instead of "a pitcher of water" and clearly identifies that it is one of the players requesting lemonade.