See the paper The Atiyah bundle and connections on a principal bundle by Indranil Biswas.
Let $p:E_G\rightarrow M$ be a principal $G$-bundle. Michael Atiyah in his paper uses a exact sequence of vector bundles over $M$, namely $$0\rightarrow \text{ad}(E_G)\rightarrow \text{At}(E_G)\rightarrow TM\rightarrow 0$$
to define a connection on the principal bundle as a section of above exact sequence. Here,
- $\text{ad}(E_G):(E_G\times\mathfrak{g})/G\rightarrow M$ is the adjoint bundle associated to the principal $G$-bundle $E_G\rightarrow M$ with adjoint action of $G$ on $P$.
- $\text{At}(E_G):TE_G/G\rightarrow M$ is the Atiyah bundle. Consider the tangent bundle $TE_G\rightarrow E_G$. Action of $G$ on $E_G$ induce an acion of $G$ on $TE_G$ as well. This map $TE_G\rightarrow E_G$ is $G$-equivariant, giving a vector bundle $TE_G/G\rightarrow E_G/G$. As $E_G/G\cong M$, this gives vector bundle $TE_G/G\rightarrow M$, a vector bundle over $M$.
The map $E_G\times \mathfrak{g}\rightarrow TE_G$ given by $(p,A)\rightarrow (\delta_p)_{*,e}(A)$ where $\delta_p:G\rightarrow E_G$ is given by $g\mapsto pg$. Even this map $E_G\times \mathfrak{g}\rightarrow TE_G$ is $G$-equivariant, giving $(E_G\times \mathfrak{g})/G\rightarrow TE_G/G$. This map of manifolds is good enough to give a morphsim $\text{ad}(E_G)\rightarrow \text{At}(E_G)$
of vector budnles over $M$. The map $P\rightarrow M$ gives $TP\rightarrow TM$. This map is $G$-invarinat, giving $TP/G\rightarrow TM$. Again, this map is good enough to give a morphism $\text{At}(P)\rightarrow TM$ of vector bundles over $M$.
Combining above morphisms of vector bundles over $M$, this gives an exact sequence of vector bundles
$$0\rightarrow \text{ad}(P)\rightarrow \text{At}(P)\rightarrow TM\rightarrow 0.$$
Then, author says, a connection is splitting of above exact sequence.
There are two ways (that I know) to think about splitting. One way is to talk about the direct sum $\text{At}(E_G)\cong \text{ad}(E_G)\oplus TM$ and the second is a section $s:TM\rightarrow \text{At}(E_G)$.
To relate to the notion of connection that we already know (as a distribution $\mathcal{H}\subset TE_G$) satisfying some properties, it is useful to think of splitting as a section $s:TM\rightarrow \text{At}(E_G)$. The map $p:E_G\rightarrow M$ pullback the bundle $\text{At}(E_G)\rightarrow M$ to give $p^*(\text{At}(E_G))\rightarrow E_G$ and pullback the bundle $TM\rightarrow M$ to give $p^*TM\rightarrow E_G$. Further, the map $s:TM\rightarrow \text{At}(E_G)$ is pullback to give $D:p^*TM\rightarrow p^*(\text{At}(E_G))$.
In page number $301$ (equation $2.7$), they defines a morphism of vector bundles $\mu:p^*(\text{At}(E_G))\rightarrow TE_G $ and observes that this is an isomorphism of vector bundles.
Consider the composition $p^*TM\xrightarrow{D} p^*(\text{At}(E_G))\xrightarrow{\mu} TE_G$.
Image of this map, i.e., $\mathcal{H}(D):=(\mu\circ D)(p^*TM)\subset TE_G$ is a sub bundle. This plays the role of Horizantal sub bundle when defining connection in the sense of Kobayashi and Nomizu.
This is how we relate the two notions of connection on a principal $G$-bundle $E_G\rightarrow M$, one as a splitting of the Atiyah sequence of vector bundles and the other as a distribution of $TE_G$ satisfying some properties.
Another way I learned from Ralph Cohen's notes (Page number $35$). He does not say in this form, I am interpreting. One can pullback the
exact sequence $$0\rightarrow \text{ad}(E_G)\rightarrow \text{At}(E_G)\rightarrow TM\rightarrow 0 $$
of vector bundles over $M$ along the map $p:E_G\rightarrow M$ to get the exact sequence
$$0\rightarrow p^*(\text{ad}(E_G))\rightarrow p^*(\text{At}(E_G))\rightarrow p^*(TM)\rightarrow 0 $$
of vector bundles over $P$. Ralph Cohen defines a connection on $p:E_G\rightarrow M$ to be a $G$-equivariant splitting of the above exact sequence of vector bundles. As mentioned above, there is an isomrophism of vector bundles
$p^*(\text{At}(E_G))\cong TE_G$, that gives following sequence $$0\rightarrow p^*(\text{ad}(E_G))\rightarrow TE_G\rightarrow p^*(TM)\rightarrow 0 $$
of vector bundles over $P$. A section of above sequence $s:p^*(TM)\rightarrow TE_G$ gives a sub bundle $\mathcal{H}:=s(p^*(TM))\subset TE_G$, giving a distribution. This is compatible with the structure of prinipal bundle, giving a connection in the sense of distribution $\mathcal{H}\subseteq TE_G$.
