Confusion about the proof of $D\phi=d\phi+\rho_*(\omega)\cdot\phi$ and $D\omega=d\omega+\frac12[\omega\wedge\omega]$ in principal bundle

Question

The question focus on the proof from Note 1 in Wikipedia, the proof is about $D\phi=d\phi+\rho_*(\omega)\cdot\phi$ but it should work for $D\omega=d\omega+\frac12[\omega\wedge\omega]$ with minor modification.

Definitions

Here $D$ is covariant derivative on principal G-bundle $P$ defined as $D\phi(v_0,...,v_k)=d\phi(hv_0,...,hv_k)$ where $h$ is the projection to horizontal subspace according to the given principal connection.

$\phi$ is a tensorial (or basic, i.e. G-equivariant and horizontal) k-form on $P$.

$\rho:G\to GL(V)$ is a representaion. $\rho_*:\mathfrak{g}\to\mathfrak{gl}(V)$ is $(d\rho)_e$.

$\omega$ is the principal connection form.

Idea

The basis idea there seems to prove $D\phi-d\phi=\rho_*(\omega)\cdot\phi$ and $D\omega-d\omega=\frac12[\omega\wedge\omega]$. To prove that, given any vector field $X_0,...,X_k$ that is to be input to the k+1-form on the left, first expand these vector fields into some "special vector fields" which have special properties that can make most terms vanish in the later process.

For example, the "special vector fields" can be horizontal and vertical vector fields. in this case, $$D\phi(X_0,...,X_k)-d\phi(X_0,...,X_k) =d\phi(hX_0,...,hX_k)-d\phi(vX_0+hX_0,...,vX_k+hX_k) =-\Sigma_i d\phi(p_{i0}X_0,...,p_{ik}X_k) \tag 1$$ , where $vX$ is the projection to the vertical subspace and $p_{ij}X$ means either $vX$ or $hX$, and the term that all $p_{ij}X$ is $hX$ is excluded.

Then apply "invariant formula for exterior derivative": $$d\phi(X_0,...,X_k)=\frac 1{k+1}\Sigma_i (-1)^i d_{X_i}\phi(X_0,...,\hat X_i,...,X_k)\\ + \frac 1{k+1}\Sigma_{i<j} (-1)^{i+j}\phi([X_i,X_j],X_0,...,\hat X_i,...,\hat X_j,...,X_k) \tag 2$$ on each term of the expansion (1) and sum the result.

But the above example of expansion doesn't work since horizontal and vertical vector fields are not "special enough", they can make most terms in (2) vanish, but not all. For example $[vX_i,hX_j]$ may not be zero which make some term in the second sum in (2) not vanishing.

But if the vector field input to (2) is more special, such as fundamental vector field (a special case of vertical vector field), more terms in (2) vanish and the result is there:

$$ D\phi(X_0,...,X_k)-d\phi(X_0,...,X_k)=\frac 1{k+1}\Sigma_i (-1)^i \rho (\omega (X_i))\phi(X_0,...,\hat X_i,...,X_k) $$

Confusion

Base on the idea section above, the confusion is about this from Note 1 of Wikipedia:

For the general case, let $X_i's$ be tangent vectors to $P$ at some point such that some of $X_i's$ are horizontal and the rest vertical. If $X_i$ is vertical, we think of it as a Lie algebra element and then identify it with the fundamental vector field generated by it. If $X_i$ is horizontal, we replace it with the horizontal lift of the vector field extending the pushforward $\pi X_i$. This way, we have extended $X_i's$ to vector fields. Note the extension is such that we have: $[X_i, X_j] = 0$ if $X_i$ is horizontal and $X_j$ is vertical

The questions are:

1. When will $[X_i,X_j]=0$ work? Does it only work when one is horizontal and the other is fundamental vector field? Will it also work for any horizontal and vertical vector field? This is answered here.

2. What is exactly the horizontal vector field defined here? What is "extending the pushforward $\pi X_i$"? To be specific, Note 1 begin with some horizontal tangent vector (not vector field) $X_i$ at some point $p$ of $P$, then it is push forward to $\pi_* X_i$ in $T_{\pi(p)}M$, then how is $\pi_* X_i$ extended to a vector field on M? (the later lift back to vector field on P seems not to be a problem)

3. Why does Note 1 use such special horizontal vector field? Doesn't any horizontal vector field work for the above idea?

4. According to the idea above, what is the expression of expasion of $X_0,...,X_k$ into the "special vector field" defined in Note 1? I'm not even sure it can be expanded with such vector field. (Note that in the above idea section, my expansion is clear and well defined, though it doesn't work for the proof)