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My name is Avigayil, and I've I see all the various nicknames as just that - nicknames. Basically if your mother named you simply Abby, it doesn't change the meaning. If you met someone named Jim or Frank, you wouldn't imagine that name having a different meaning than James or Francis. The same rule applies to girls. A nickname is always a nickname, in the meaning sense. At least that's what I believe, as a proud holder of the name.
If this helps any, my name used to be spelled the traditional way (Abigail) and I always went by Abi. But I never tried changing the meaning of Abi just because I absolutely never went by Abigail. I feel the same rule partially applies to you: Your real name is not Abigail, and I never saw it as mine either, which is part of the reason I made a few alterations. To help make your point, when I switched to Avigayil, I decided to go by Ava instead of Avi. Ava has its own separate meaning from Avigayil, and I tend to see it as just a dual-meaning. I don't think it's really a big deal.
But in all honesty, this is YOUR name, not society's. So while I feel the meanings are the same, you can interpret your own name however you want. =)

Since the name Abby is derived from Abigail I would venture to say it does have the same meaning as Abigail, just like Jenny has the same meaning as Jennifer or Tim as Timothy.