Lark, from personal experience, I can tell you that you're *mostly* wrong. Pets CAN be familiars. My familiar is a pet, and yet she helps me in my magic works in the Earth Element and has traveled with me in the Astral. She's even visited me in the Dreamtime.
I have five pet snakes, But yet only one of them is my familiar. I have three cats as well, but they are not my familiars.
I did not even buy her, really. I won her in a game when I bought a different snake of mine.