I don't think this is a form of reproduction. I believe it is the result of an injury. In over 20 years I've never seen evidence of a bud like this producing a separate anemone. At least not with haddoni. I've heard several stories like Gary's above, though. I don't believe this bud will ever form a pedal disk and be able to survive on its own.
Elegance corals do something similar. If the oral disk is punctured, when it heals, it often forms a ring of tentacles around the wound. This looks like a "bud" or small elegance growing off of the coral.