False or not

Some years ago I bought a crab cactus/false Christmas cactus/November cactus in November. It had pale pink flowers.Following year it bloomed at Easter, then late in the summer and in November again and so it has continued. The cactus changed the colour of the flowers, to white. Now it is blooming again, just for Christmas! One of the names "false Christmas cactus" is not right anymore!