I feel like the kid that blurts out "There is no Santa Claus" to all the other children.

Anyone can call their cactus anything they want, of course. There is a Schlumbergera cactus that starts blooming at Christmas time, not the month leading up to Christmas. We are being short-changed by the commercial marketers to find only the species that blooms in the month leading up to Christmas, when it's easier to sell plants with flowers on them. The traditional, real Christmas cactus is becoming hard to find and nobody even appreciates that. Someone who has the traditional Christmas cactus has something special. That cactus is Schlumbergera bridgesii, or S. russelliana, or Schlumbergera x buckleyi - the botanical names have changed over the years.

But the plants in both these posts - and my own post - they are all S. truncata. They are pretty. They come in a lot of different colors. And they bloom in the month leading up to Christmas.

Here are a few references that talk about the differences:
Horticulture Magazine
Cooperative Extension at Clemson University
The Gardener, Cook Blog.

If it sounds like I'm on some sort of high horse, check out that last blog. That guy is really upset about it, lol.