The Christofel symbol can be thought of as describing how a coordinate system twists and stretches as you move about on the manifold. It thus is about a particular coordinate system in two different ways:
It is possible to express the twisting and stretching of one coordinate system in another coordinate system. If we keep the latter coordinate system unchanged then that is a tensor expression referring to some subject coordinate system.
This is the source, I think, of the confusion in this discussion of the covariant derivative.