Maybe the change of the article depends upon our certainty of the object's origin. In the case of the apple, we are sure that the apple mush was at one point an entire apple. We saw the original apple. It was smashed apart. The chunks on the counter, the juice, the mush. They are apple. If we came to the scene after the smashing, would it still be apple? There would be a period of recognition. The taste, the smell, the look. Maybe we could identify it as apple. If we can't then it's something else all together.
I disagree with the example. If you had a recipe for apple cake that called for "one mashed apple," for example, you would not think twice if the instructions said to "add the apple to the batter." By the same token, if it called for "three mashed apples,"then said to "add the apples to the batter," you would understand very clearly.
This doesn't undercut the example at all. One uses the article because of the recipe context which calls for specific quantities of ingredients. In the example sentences you have given, what is being being asked is this: "Add the "substance which we have come to call 'apple'" to the batter (not "the apple" meaning a whole apple). It is the difference between saying "the apple" and the "apple". If you follow.
Going back to Cristi's example. When confronted with the apple on the counter one could say "who got apple all over the counter?" However, if that person were commanding someone else to clean the "apple" off of the counter than they would say "clean the apple off of the counter". The same grammatical tricks are at play here as in the recipe example. Cristi?