A dog jumping for a Frisbee in the snow.
A pet is enjoying a game of fetch with his owner.
The pet can also play a game in the snow.
A dog need not be a pet.  A dog jumping for a Frisbee need niether be participating in willful play, nor enjoying it.  Anyone can toss a Frisbee in the direction of an animal and need not be its owner.
Not all dogs have an owner.