I think you caught me there on a verbiage technicality.
I probably should have worded it better in the additional details of the premise, but my intent was that your old self simply doesn't exist in this timeline. Your family is there, but they had some other kid instead of you on the same date you were born.
If your name is Jarnathan Smith, you like baseball and your favorite color is puce, you'd instead find that your family had a kid named Archideld Smith who likes rugby and whose favorite color is mauve.
This may be one of the better ideas I've read so far. Incredible, kudos!