So why bother with me? Seems that you're unwilling to entertain the possibility that you made an error. In any case Eq. #2 in my page is the transformation properties of the components of a contravariant vector.
However I see your point in Dirac. It was confusing since he never named them. Dirac and Ramond, two authors you quote, use different definitions of the Christoffel symbols of the second kind.
However it appears that you're having a bad day since I find your comments to be irritating.