Actually Bob is using a well known theorem:
Theorem. If $L: V \to V$ is a linear map from a finite dimensional vector space to the same space, there is a uniquely determined tensor $T \in V \otimes V^*$ such that induces $L$ as follows. If $v\in V$, $Lv$ is the contraction of the tensor product of $T$ and $v$ (the only possible contraction).
In our case, $V= T^*_pM$, with $p$ fixed, $L$ is the difference of the two covariant derivatives evaluated at $p$. Your second statement assures that $L$ is well defined as it, differently form each covariant derivative, depends on the value of its argument at $p$ and not on what happens in a neighborhood of $p$. Thus, the difference of the two derivatives at $p$ is a linear map in $T^*_pM$ with $p$ fixed. The tensor corresponding to that map in view of the quoted theorem is the mentioned tensor $C(p)$. Varying $p$ in $M$ you define a tensor field.