Different notions of connection? I am trying to connect two different notions of connection I learned:
I first learned the notion of connections in Demailly's "Complex Analytic and Differential Geometry", where in the section called "Linear Connections", he defined a connection $D$ on a bundle $E$ as

a linear differential operator of order 1 acting on $\mathcal{C}_*^\infty(M,E)$ and satisfying the following properties:
$$\text{(i)} D:\mathcal{C}_q^\infty(M,E)\to \mathcal{C}_{q+1}^\infty(M,E)$$
$$\text{(ii)} D(f\wedge s)=df\wedge s+(-1)^q f\wedge Ds$$ for any $f\in \mathcal{C}_p^\infty(M,K)$ and $s\in \mathcal{C}_q^\infty(M,E)$.

Here he appears to use $\mathcal{C}_q^\infty(M,E)$ to denote the bundle $\bigwedge^q T^*M\otimes E$
At the start of "Elliptic operators, topology and asymptotic methods", the author defines a connection on a vector bundle $V$ as

a linear map $$\nabla: C^\infty(TM)\otimes C^\infty(V)\to C^\infty(V)$$ assigning to a vector field $X$ and a section $Y$ of $V$ a new vector field $\nabla_X Y$ such that, for any smooth function $f$ on $M$,
$$\text{(i) }\nabla_{fX} Y=f\nabla_X Y$$
$$\text{(ii) }\nabla_X(fY)=f\nabla_X Y+(X.f)Y,$$ where $X.f$ denotes the Lie derivatives of $f$ along $X$.

Are these two notions of connection the same? Would $\nabla_X\nabla_Y(s)$ be the same as $D^2(s)$ evaluated on $(X,Y)$? (I am not sure how to evaluate a section of $\bigwedge^2 T^*M\otimes E$ on a pair of vector fields, so this is my best guess of how could these two be equivalent.)
 A: The first definition sounds much more like the exterior covariant derivative associated to a linear connection (as in the second definition); another common notation for it is $d^{\nabla}$ or $d_{\nabla}$. Also, in the second definition, you write

assigning to a vector field $X$ and a section $Y$ of $V$ a vector field $\nabla_XY$...

the second instance of vector field (emphasis mine) is a typo, it should say "a section $\nabla_XY$ of $V$".
Also, in the first definition, $D^2s$, the second covariant exterior differential of a smooth section $E$-valued $q$ form $s$ on $M$, equals the 'wedge' of the curvature, $R$, of the connection (which is a smooth $\text{End}(E)$-valued $2$-form on $M$, i.e $R\in \mathcal{C}^{\infty}_2(M,\text{End}(E))$) and $s$:
\begin{align}
D^2s&=R\wedge_{\epsilon}s.
\end{align}
Here, $\wedge_{\epsilon}$ denotes the wedge product with respect to evaluation of endomorphisms on vectors: see Ivo Terek's answer here for details. Its value on a pair of vector fields $X,Y$ is
\begin{align}
(D^2s)(X,Y)&=\nabla_X\nabla_Ys-\nabla_Y\nabla_Xs-\nabla_{[X,Y]}s.
\end{align}

As far as terminology is concerned, I find this usage confusing, but the operators are essentially the same when acting on sections of $E$, i.e $D:\mathcal{C}^{\infty}_0(M,E)\to \mathcal{C}^{\infty}_1(M,E)$ and $\nabla:C^{\infty}(TM)\otimes C^{\infty}(E)\to C^{\infty}(E)$ are related as:
\begin{align}
(Ds)(X)&=\nabla_Xs.
\end{align}
Or, $Ds=\nabla s$, where it is understood that there is an open slot to be filled in by a vector field $(Ds)(\cdot)=\nabla_{(\cdot)}s$; this open slot for a vector field is what gives the 1-form nature.
