The Pilgrim in the Pine Forest
The definitive answer was given on BoardGameGeek by Hanno Girke (editor of the rules).

The Pilgrim has priority.
The Pine Forest just states a standard rule, it doesn't create a new rule.
Therefore, as soon as you play Pilgrim, you override this rule.

There is some background at play-agricola.com, where the Gamer's Deck was designed.
As noted by Hanno, the standard rule is on the Pine Forest.  You may not give goods from your supply to the general supply.