Regarding the final version of John's promise, we find in Chapter 5 after Mrs. Dashwood announces her intention to move to Barton Cottage:

"He really felt conscientiously vexed on the occasion; for the very exertion to which he had limited the performance of his promise to his father was by this arrangement rendered impracticable."

He had assuaged any guilt he might have felt at not giving them money by determining that an occasional gift of game or produce would be a sufficient exercise of his duty. But now that they were moving so far away, that would be impossible. He would now have no way of convincing himself that he had done all that was necessary for them. Fanny might have been able to easily persuade him to do so little, but I think this short passage shows that it still nagged at him in the back of his mind.