I've often seen it written that Christmas Cactuses (Shlumbergera) are not "true" cacti. But I understand that they are genetically of the cactus family? What is a "true" cactus?
It is a true cactus, but a jungle cactus rather than a desert cactus. Check out this link to learn more…http://www.cactiguide.com/growcacti/
Have you ever seen a plant called Pereskia? It has woody stems and leaves yet is in the family Cactaceae so is a cactus. Genetics calls the shots.