A 'different kind of Christmas story' might involve a group of animals who find a way to exchange gifts in the forest. For example, squirrels sharing their nuts, birds offering their feathers for decoration, and deer using their antlers to carry presents. They create their own Christmas traditions without any human interference.
Well, it could include non - traditional characters like aliens celebrating Christmas on Earth. Maybe they learn about the concept of giving and sharing from humans and then have their own unique way of doing it.