Eheresmann connection and covariant derivative

Question

I was introduced the following definition of connection on a fibre bundle (actually my professor gave the same definition in the context of fibred manifolds). Which I understood is due to Ehresmann

A connection on a fibre bundle $\pi:B\rightarrow M$ is a distribution $H\hookrightarrow TB$ such that: $$\forall b\in B\quad T_bB=H_b\oplus V_b(\pi)$$ where $V_b(\pi)\subset TB$ is the subspace of vertical vectors at $b\in B$: $$V_bB=\{v\in T_bB| T_b\pi(v)=0\in T_{\pi(b)}(M)\}$$

We can identify a connection with a map $\omega: TM\rightarrow TB$ as follows

Fix $H$, $u\in T_bB$, $v=T_b\pi(u)\in T_{\pi(b)}M$. Then $u\equiv u_{(H)}\oplus u_{(V)}$. Call $\tilde{v}:=u_{(H)}$ the horizontal lift of $v$, this is uniquely defined: $u'\in(T_b\pi)^{-1}(v)\Rightarrow T_b\pi(u-u')=v-v=0 \Rightarrow u-u'\in V_b(\pi)\Rightarrow u_{(H)}=u'_{(H)}$
Consequently a connection defines $\omega_H:TM\rightarrow TB$, fiberwise: $$\forall b\in B,\ (\omega_H)_b:T_{\pi(b)}M\rightarrow T_bB\quad\ v\in T_{\pi(b)}M\mapsto \tilde{v}\in T_bB$$ such that $T_b\pi\circ\omega_H=id_{TM}$.
Conversely,any such map determines a connection $H_\omega$, by setting $$(H_\omega)_b =\{u\in T_bB:\ u\in Im(\omega_b)\}$$

Moreover $\omega$ makes the following Witney sequence split: $$0\longrightarrow VB(\pi)\longrightarrow TB\xrightarrow{T\pi}TM\longrightarrow 0$$

Now, my professor goes on defining a notion of covariant derivative for these generalized connections as follows

The covariant derivative of a connection $H$ on a bundle $\pi:B\rightarrow M$ is the mapping: $$\nabla:\chi(M)\times\Gamma(B)\rightarrow\Gamma(VB)\quad\nabla_\xi\sigma=T(\sigma)(\xi)-\xi_H\circ\sigma$$
where $\Gamma(B)$ are the sections of $B$, $\chi(M)$ vector fields on $M$, $T$ the tangent map and $\xi_H$ the horizontal lift of a vector field $\xi$. So, $\nabla_\xi\sigma$ is a vertical vector field over the section $\sigma$, in the sense that $\nabla_\xi\sigma$ is a section of $M→VB$ and considering the projection $τ:VB→B$ we obtain a section of $M→B$, $\tau\circ\nabla_xi\sigma$

1. Now, what properties does this operator have? I couldn't find its definition elsewhere than in my notes: on the contrary Wikipedia (https://en.wikipedia.org/wiki/Ehresmann_connection) seems to motivate the generalized definition of Eheresmann connection to drop the covariant derivative.
2. In particular, in the case of a vector bundle it is known that the Eheresmann definition of linear connection and the Koszul one (which is in term of an operator as above) are equivalent. But does it hold generally that the operator defined above completely identifies a connection in the general context of fibre bundles?