What's this use of the comma called?

Question

Each group will receive one pie and one apple per child.

Each group will receive one pie, and one apple per child.

In the first case each group receives many apples and many pies. In the second case each group receives many apples, but each group receives only one pie.

I need to describe the rule that makes it so the comma changes the meaning in this sentence, but I'm not sure what the rule operating here is called, nor what the relationship between these two sentences is called.

(While this is about the use of the comma, I strongly suspect the comma here is delineating two different syntactic reasonings resulting in the same words in the same order, so the grammar tag has been included. If you object to this use of the grammar tag, you can remove it. Just at least think about it first :P )

Answer 1

The issue with the sentence is syntactic ambiguity, but it is only weakly related to the use of the comma.

Syntactic Ambiguity ... in phrase reference
"I saw him on the hill with a telescope"; ....
The first sentence has three readings which depend on how the last prepositional phrase is interpreted. If 'with a telescope' is read as accompaniment, it can modify 'him' or 'hill' and produce "he and a telescope were seen on the hill" or "he was seen on the hill which has a telescope on it". If the phrase is taken as identifying the instrument used for seeing, it produces "he was seen on the hill by use of a telescope".
- U. Ottawa

Your example can be treated as a form of ellipsis. The ambiguity arises from which words are deemed to have been elided:

1. Each group will receive one pie(,) and each group will receive one apple per child.
2. Each group will receive one pie per child(,) and one apple per child.

Interpretation #1 gives the group a single pie while interpretation #2 supplies each child with their own pie. Admittedly, where the comma is present in the original, it tends to distance pie from child, favouring interpretation #1.