A connection $A$ on $P$ can be viewed in many ways:
A $G$-invariant 'horizontal' subbundle $H_A \subset T P$ transversal to the vertical tangent space $V T P=\operatorname{ker}(d \pi)$;
a system of parallel transport in $P$ (lifts of paths in $X$ to horizontal paths in $P$ );
a $\mathfrak{g}$ valued, ad-equivariant 1 form equal to $i_p^{-1}: V T P_p \rightarrow \mathfrak{g}$ (where $i_p$ is the differential of the action of $G$ on $P$ );
a covariant derivative $\nabla_A: \mathcal{C}^{\infty}(X ; E) \rightarrow \mathcal{C}^{\infty}\left(Y ; T^* X \otimes E\right)$ satisfying a Leibniz rule $\nabla_A(f \sigma)=f \nabla_A(\sigma)+d f \otimes \sigma$.
I have seen the rough equivalence of the first three notions, but am not sure how to relate these to the last one.