The reason for that discrepancy between dog and cop is of course that there is no such thing as a "non-action", it is too wishy washy terminology that doesn't help form an understanding of the details of the rules.
The cop is a free triggered ability. You don't use those just anywhere, but in specific timing windows. You don't have such a timing window between assigning and application of damage.
The dog is a reaction triggered ability. You use those in response to listed triggering conditions. The dog's specific condition does allow you to trigger it between assigning ("dealing") and actual application of damage. (This is probably mostly due to the dog using "when" instead of "after", which means it interrupts the triggering incident)