Let $E\to M$ be a vector bundle above a manifold $M$, with a connection $\nabla$ defined on the tangent bundle, and let $\nabla^{E}$ be a linear connection on $E$ and $\omega$ a $n$-form on $M$ with valued in $E$ i.e. an element of $\Omega^n(M, E)$.
From different sources, I gather the following definitions and a result (using usual notations)
The exterior covariant derivative of $\omega$ is an element of $\Omega^{n+1}(M, E)$ defined by $$d^{E}\omega(X_{0}, \dots, X_{n})=\sum_{i=0}^n(-1)^{i}\nabla_{X_{i}}^{E}(\omega(X_{0}, \dots,\hat{X}_{i}, \dots, X_{n}))\\ +\sum_{i<j}(-1)^{i+j}\omega([X_{i},X_{j}], X_{0}, \dots, \hat{X}_{i}, \dots, \hat{X}_{j}, \dots, X_{n})$$
The covariant derivative of $\omega$ is an element of $\Omega^1(M)\otimes\Omega^n(M, E)$, and not of $\Omega^{n+1}(M, E)$, defined by $$(\nabla_{X}^{E}\omega)(X_{1},\dots, X_n)=\nabla_X^{E}.(\omega((X_{1},\dots, X_n)) -\sum_{i=1}^n \omega(X_{1},\dots, \nabla_{X}X_{i},\dots, X_n)$$
When the connection on $M$ is symmetric, i.e. without torsion, then the two are connected by $$d^{E}\omega(X_{0}, \dots, X_{n})=\sum_{i=1}^n(-1)^{i}\nabla_{X_i}^{E}\omega(X_0, \dots, \hat{X}_i \dots, X_n)$$
The proof of 3 seems easy enough by unravelling the definitions, but I was unable to find clear references that do not use local coordinates for definition 2 and formula 3 proof. Do they make sense and where can I find them?
Actually by digging around more deeply I found this quite interesting and detailed answer of Ribeiro to an MO question Ribeiro_answer that generalizes this result to higher order iteration of the derivatives and the connections, with or without torsion.