Lately, I have been wondering about how to make the reasoning of uniqueness of the Levi-Civita connection $\nabla$ on a tangent bundle $\pi:TM\to M$ with a bundle metric $g$, via f.e. the Koszul formula, compatible with the interpretation of (Ehresmann) connections as horizontal distributions $\mathscr{H}$ of $TTM$ (such that $\mathscr{H}\oplus ker(d\pi)=TTM$). By the latter, I mean the definition where a covariant derivative of a section $s\in\Gamma (TM)=\mathscr{X}(M)$ is given by
\begin{equation} \nabla_Xs := (\pi^\mathcal{\nabla}\circ Ts)(X), \end{equation}
where $\pi^{\nabla}: TTM\to ker(d\pi)$ denotes the projection onto the vertical component. In how far does the choice of a metric (and the assumption of torsion-free) naturally fix such a horizontal distribution $\mathscr{H}$? Originally, I thought that given a metric, we have a natural choice of horizontal complements by taking the $g-orthogonal$ complements in each fiber. But it seems I mixed up orthogonality on the level of $TM$ with the level of $TTM$. Then I thought about considering an induced metric like $dg$ on $TTM$, but that does not really amount to a metric on $TTM$, or does it?
My main question boils down to asking for an interpretation, visualization or argument of the uniqueness of the LC connection in this Ehresmann picture of horizontal distributions. (Any comment on the differential map $dg$ of a metric $g$ would also be much appreciated.)
Thank you for taking the time and interest.
Edit: Thank you, @J.V.Gaiter for pointing out that torsion and hence also the LC Connection only make sense on a tangent bundle (and associates) not just any vector bundle $E$.