I can think of one good reason to separate yourself from your real name if you are a professional dancer and that is potential stalkers. I don't use my real name online to keep potential problems away. I can certainly see why someone else might do the same (including deliberately looking different) for a dance career.
lily_raks, is the second picture a Bella? Looks like one
If you worked in a conservative field, would you change your look (wear a wig per say) to not be easily recognized by your peers or boss?
Most dancers do that and for good reason: pervs are everywhere. I used to dance under my full name when I was a starting pro dancer and that was before it became popular to do that (like Rachel Brice does), but I soon stopped it, it caused too much confusion (my family name is Jewish, Arabic and Turkish LOL) and safety issues.
And a story about being recognized:
(I can't find the horrible picture of me, the dress was double my size, I had socks on and ballet shoes) I had a gig a block away from home, so I put on a dress, wig and head scarf. Thought it would be very hard to recognize me later (I was blond back then). And some time later a kid screamed at me in the street: 'belly dancer!'
So it doesn't always work...