The question concerns the partial covariant differentiation of a relative scalar of weight $-1$, in David Lovelock and Hanno Rund's Tensors, Differential Forms, and Variational Principles, Dover publication, 1989.
On page 107, a general formula for the partial covariant differentiation of a relative tensor $\Lambda^{j_1\cdots j_r}{}_{l_1\cdots l_s}$ with weight $w$ is given by: $$ \begin{aligned} \Lambda^{j_1\cdots j_r}{}_{l_1\cdots l_s|k}=\dfrac{\partial\Lambda^{j_1\cdots j_r}{}_{l_1\cdots l_s}}{\partial x^k}+\sum_{\alpha=1}^r\Gamma_m{}^{j_\alpha}{}_k\Lambda^{j_1\cdots j_{\alpha-1}mj_{\alpha+1}\cdots j_r}{}_{l_1\cdots l_s}\\-\sum_{\beta=1}^s\Gamma_{l_\beta}{}^m{}_k\Lambda^{j_1\cdots j_r}{}_{l_1\cdots l_{\beta-1}ml_{\beta+1}\cdots l_s}-w\Gamma_k{}^h{}_h\Lambda^{j_1\cdots j_r}{}_{l_1\cdots l_s} \end{aligned}\tag{1.24} $$ Notice the additional term with contracted Christoffel symbol, which does not exist for a tensor.
A specialized version of the above equation appears, on page 168, for a relative scalar $X$ with weight $-1$: $$ X_{|k}=\dfrac{\partial X}{\partial x^k}+X\Gamma_l{}^l{}_k=\cdots $$ Notice, however, that this time, the contracted Christoffel symbol is over a different pair of contra-/covariant indices. At this point, symmetry of the connection is not assumed in the text (it was only in the second paragraph on page 169 that symmetry of the connection is mentioned).
So, my question is: is the second equation shown above correct? Or the author did actually use symmetry of the connection without realizing it?