What about a square that contains just a sliver of the car from the next box over? Does that still count? I hypothesize that my attempt to classify every pixel related to a car as a "car" may contribute to my failing of these tests.

This is what kurtisc is saying. The algorithm is unable to classify those pixels, so you have to guess which bits of the car can be detected by the algorithm and then select only those. So you have machines asking humans to think like a machine in order to prove they are human.