In the book Foundations of Differential Geometry by Kobayashi & Nomizu, the fundamental vector field is defined in section 4, pg.42:

If $G$ acts on $M$ on the right, we assign to each element $X\in G'$ a vector field $X^*$ [called the fundamental vector field] on $M$ as follows. The action of the 1-parameter subgroup $a_t:=exp(tX)$ on $M$ induces a vector field $X*$ on $M$.

Where we have denoted $G'$ as the Lie algebra of $G$. Also the following proposition 4.2 states using the same setup

The mapping $\sigma: G' \rightarrow X(M) $ which sends $X$ into $X^*$ is a Lie algebra morphism. If $G$ acts effectively on $M$, then $\sigma$ is an isomorphism of $G'$ into $X(M)$. If $G$ acts freely on $M$, then, for each non-zero $X\in G'$, $\sigma(X)$ never vanishes on $M$.

Then in section 5, where $P\rightarrow M$ is a principal bundle with structure group $G$ they write:

Since the action of $G$ sends each fibre into itself, $(X^*)_p$ is tangent to the fibre at each $p\in P$. As $G$ acts freely on $P$ [by definition of a principal bundle], $X^*$ never vanishes on $P$ (if $X!=0$) by proposition 4.1. The dimension of each fibre being equal to that of $G'$, the mapping $X\rightarrow (X^*)_p$ of $G'$ into $T_pP$ is a linear isomorphism of $G'$ onto the tangent space at $p$ of the fibre through $p$.

Also, on the book Natural Operations in Differential Geometry by Michor, Kolar & Slovak which is freely available on the net they define the fundamental field and establish some of its properties in lemma 5.12 and then in theorem 10.18 they write:

The vertical bundle $VP\rightarrow P$ of the principal bundle $P\rightarrow M$ is trivial as a vector bundle over $P$; hence $VP =~ P \times G'$

Then in the following section they show how to obtain a connection form $\omega$ from any connection on $P$; the form $\omega$ is a 1-form on $P$ valued in $G'$ and they relate this in a lemma to the fundamental field:

$\omega$ reproduces the generators of the fundamental fields: $\omega(X^*_p) = X$

Finally, Nlab defines the dual notion of a fundamental form:

Given a Lie group $G$ with a right action on a manifold $M$ then a fundamental 1-form is a 1-form valued in $G'$, that is $\omega \in \Gamma(T*M \otimes G')$, such that $\omega_p(X^*_p)=X$

In this language, a connection 1-form is simply a fundamental 1-form as Michor Lemma 11.1 points out; and it's a principal connection iff it is $G$ equivariant, that is:

$((r^g)\omega)(X_p) = Ad(g^{-1}.\omega(X_p)$

In the book Foundations of Differential Geometry by Kobayashi & Nomizu, the fundamental vector field is defined in section 4, pg.42:

If $G$ acts on $M$ on the right, we assign to each element $X\in G'$ a vector field $X^*$ [called the fundamental vector field] on $M$ as follows. The action of the 1-parameter subgroup $a_t:=exp(tX)$ on $M$ induces a vector field $X^*$ on $M$.

Where we have denoted $G'$ as the Lie algebra of $G$. Also the following proposition 4.2 states using the same setup

The mapping $\sigma: G' \rightarrow X(M) $ which sends $X$ into $X^*$ is a Lie algebra morphism. If $G$ acts effectively on $M$, then $\sigma$ is an isomorphism of $G'$ into $X(M)$. If $G$ acts freely on $M$, then, for each non-zero $X\in G'$, $\sigma(X)$ never vanishes on $M$.

Then in section 5, where $P\rightarrow M$ is a principal bundle with structure group $G$ they write:

Since the action of $G$ sends each fibre into itself, $(X^*)_p$ is tangent to the fibre at each $p\in P$. As $G$ acts freely on $P$ [by definition of a principal bundle], $X^*$ never vanishes on $P$ (if $X!=0$) by proposition 4.1. The dimension of each fibre being equal to that of $G'$, the mapping $X\rightarrow (X^*)_p$ of $G'$ into $T_pP$ is a linear isomorphism of $G'$ onto the tangent space at $p$ of the fibre through $p$.

Also, on the book Natural Operations in Differential Geometry by Michor, Kolar & Slovak which is freely available on the net they define the fundamental field and establish some of its properties in lemma 5.12 and then in theorem 10.18 they write:

The vertical bundle $VP\rightarrow P$ of the principal bundle $P\rightarrow M$ is trivial as a vector bundle over $P$; hence $VP =~ P \times G'$

Then in the following section they show how to obtain a connection form $\omega$ from any connection on $P$; the form $\omega$ is a 1-form on $P$ valued in $G'$ and they relate this in a lemma to the fundamental field:

$\omega$ reproduces the generators of the fundamental fields: $\omega(X^*_p) = X$

Finally, Nlab defines the dual notion of a fundamental form:

Given a Lie group $G$ with a right action on a manifold $M$ then a fundamental 1-form is a 1-form valued in $G'$, that is $\omega \in \Gamma(T^*M \otimes G')$, such that $\omega_p(X^*_p)=X$

In this language, a connection 1-form is simply a fundamental 1-form as Michor Lemma 11.1 points out; and it's a principal connection iff it is $G$ equivariant, that is:

$((r^g)\omega)(X_p) = Ad(g^{-1}.\omega(X_p)$