Your definition is a sort of hybrid of two standard definitions of connection 1-form. These are:

- A
*connection 1-form* is a Lie-algebra valued 1-form $\omega$ which satisfies $\omega(A^\#) = A$ for all $A \in \mathfrak{g}$ and $ad_g(R_g^*\omega) = \omega$.
- Let $VP$ denote the vertical bundle of $P$, i.e. the kernel of $d\pi: TP \to TM$. An
*Ehresmann connection* is a complementary subbundle of $TP$, i.e. a subbundle $HP$ of $TP$ such that $TP = VP \oplus HP$ as vector bundles over $P$, which is invariant under the $G$-action: $R_g^* H_u P = H_{ug} P$.

There is a one-to-one correspondence between connection 1-forms and Ehresmann connections as follows. Given a connection 1-form $\omega$, the kernel of $\omega$ is $G$-invariant and complementary to $VP$, so it defines an Ehresmann connection. Conversely, given an Ehresmann connection $TP = HP \oplus VP$, let $q: TP \to VP$ denote the natural projection map (a morphism of vector bundles). Note that the vector fields $A^\#$ define an isomorphism between $VP$ and the trivial bundle $P \times \mathfrak{g} \to P$, so composing $q$ with the projection $VP \cong P \times \mathfrak{g} \to \mathfrak{g}$ gives a map $TP \to \mathfrak{g}$, i.e. a $\mathfrak{g}$-valued 1-form. This is a connection 1-form.

Unwinding these identifications, you could just as well use a third definition:

- A connection on a principal bundle is a Lie-algebra valued 1-form $\omega$ which satisfies $\omega(A^\#) = A$ for all $A \in \mathfrak{g}$ and whose kernel $H = \ker \omega$ is $G$-invariant.

This is your definition. From here you can deduce that $H$ complements $VP$ in $TP$, (or that $\omega$ is compatible with the adjoint action of $G$) and hence connections in this sense are the same as the objects defined in the previous two definitions.