My service dog's name is Vixen. I chose her name because she's beautiful, has a mischievous streak, and I got her in December.
She had a different name when I took her in as a rescue: "Mamas"--and shortly after I got her, my mother came for an extended visit. The poor girl was soooo confused by not knowing who I was talking to unless I was looking directly at her that I had no choice but to rename her (plus, I didn't particularly care for her original name). I noticed within days of taking her in that she had a very unique temperament, and after having her for about 2 months so we could bond, began working with her to train her as my service dog. Luckily, Vixen wanted to work for me & with me, she's extremely eager to please, highly intelligent, and a total sweetheart.
She doesn't have a different name when she's being corrected--however, I seldom use her name when correcting her. A simple 'No', or tongue click is all the correction she needs. I will use her name during recall on the rare occasions she's not by my side, and she comes running. I also call her baby, honey, darling, pretty girl, sweet girl, and my little wiggle worm--- my little wiggle worm is reserved for those few occasions when we're separated for a bit, and she's doing her happy dance to see me as we're reunited.