So I am confused: in the original setting with coordinates we have $$\nabla_i T_j = \partial_i T_j - \Gamma^k_{ij} T_k$$ Does that mean, for $\vec X_j=\partial_j \vec R$, we have $$\nabla_i \vec X_j = \partial_i \vec X_j - \Gamma^k_{ij} \vec X_k = \Gamma^k_{ij} \vec X_k - \Gamma^k_{ij} \vec X_k= 0$$
But in the conventional system we have $$\nabla_{\frac {\partial} {\partial X^i}} \frac {\partial} {\partial X^j} = \Gamma_{ij} ^k \frac {\partial} {\partial X^k} $$ What did I miss?