If its Mammillaria, at that size, it should bloom. That will answer a lot of questions really fast (the most important being 'what it is?')
I grow orchids (and cactus). You would be amazed at how fast names change in the orchid world with DNA testing and nomenclature aligning. I decided a long time ago to make note of name changes but not change the tag on my plants. I would give my plants identity crisises if I changed the tag everytime a new name came along.
