I really enjoyed reading this, @SilverBear! 
I am definitely guilty of calling pets familiars and using the terms interchangeably at times (oops! 
) Thank you very much for bringing this to light, and for explaining that there is a difference! 
For the witches that do consider their pet to be their familiar- perhaps different paths/practices have different standards or roles for what makes a “familiar”? Perhaps in some practices, what is called a “familiar” might be considered to be a “partner” or just “pet” for others?
I’m digging through my scrambled knowledge of various pagan and magickal practices to see what I can find. I don’t think that (and please correct me if I’m wrong!) familiars appear in traditional Japanese legends, although the gods could be animals or take on animal forms (like the Ainu wolf god Horkew Kamuy), and it is common for nature spirits to appear as animals and have magickal abilities in Japanese folklore and Shintoism. I suppose in this sense, the Yokai or Ayakashi spirits would be more like the small animals/imps used as familiars in Western Demonology- although I don’t know if these spirits could be entered into a contract with.
Tanuki are often portrayed as being trickster Yokai spirits with some very unique abilities! Pictures from the Wikipedia pages on Tanuki and Asian Raccoon Dogs
There are many fascinating Japanese folk legends about tanuki (Japanese raccoon dogs) and kitsune (foxes) that are real animals that can, in some cases, be kept as pets but are associated with having magickal powers and abilities. I wonder in such case, if they would be considered to be pets, familiars, or both?
I wonder in what other practices familiars appear in some shape or form? 
@MeganB, do you know if familiars are common or appear in Celtic paganism at all? And for anyone familiar with these topics (maybe @Francisco and @Abs53?) do you know if familiars play a role in Greek or Egyptian mythology? Or any other practices? 
Thank you so much, as always, @SilverBear for sharing your wisdom and opening a discussion about such an interesting topic! 
