I realize this may not be a particularly satisfying answer, but I think you're overthinking this. Remember this is a young child's homework problem. While it may not have been expressed particularly clearly, the goal is plainly not to find a single definitive answer, but to fill out the boxes using an identifiable consistent pattern.

As you surmised, Child 1 made the rows into addition problems. Child 2 made each column have a shared difference. Child 3 made the sum of the top row equal the sum of bottom row (with an arbitrary sum!). This is intended to show the test-taker what kind of answer they are seeking --not a logically unique answer, but a defensible one.

Given the numbers chosen, I would surmise that they wanted to give the test-taker an easy possibility for another possible pattern --to make each row a multiplication problem.

I realize this may not be a particularly satisfying answer, but I think you're overthinking this. Remember this is a young child's homework problem. While it may not have been expressed particularly clearly, the goal is plainly not to find a single definitive answer, but to fill out the boxes using an identifiable consistent pattern.

As you surmised, Child 1 made the rows into addition problems. Child 2 made each column have a shared difference. Child 3 made the sum of the top row equal the sum of bottom row (with an arbitrary sum!). This is intended to show the test-taker what kind of answer they are seeking --not a logically unique answer, but a defensible one.

Given the numbers chosen, I would surmise that they wanted to give the test-taker an easy possibility for another possible pattern --to make each row a multiplication problem. (2 x 26 = 52; 11 x 4 = 44). You could also give them all a common product with 208 and 88.