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The first definition is the Ehresmann connection that defines a connection on a manifold as a distribution of vector spaces completing the vertical space in the tangent space of the total space at each point.

The second definition defines a connection on a manifold as a covariant derivative, i.e. a map

∇:Γ(E)→Γ(T∗M⊗E)

where π:E→M is a vector bundle and there is a version of the Leibnitz rule as follows:

∇X(fs)=df⊗s+f⋅∇Xs

for any section s and f∈C∞(M).

I tried to write things in a chart to find out how the covariant derivation is induced from a given connection but I couldn't proceed forward. I think I haven't understood the definitions well. Would someone clarify how a distributional connection give us a connection one-form and how we can recover the horizontal space if we have a connection one-form?