[zanydroid]

This sounds like a GEC vs EGC thing. IE, the problem from colloquially calling too many things called ground & ground wire.

GEC = goes down to the ground rod.
EGC = most other "ground" thing. Definitely applies to your box to box thing.

EGC needs to be bigger since it has the job of clearing faults, so needs to carry enough current to trip the breaker.

GEC does not carry fault current in that sense.

Sometimes you there are things called grounds that are combination of GEC/EGC so you need to take the strictest size between the two, used to come up in solar circuits.