I am confused about the connection between vector bundles and locally free sheaves over schemes. In a past lecture we followed Andreas Gathmann: Algebraic Geometry (2002/2003), the definition of locally free vector sheaf and vector bundle from that source are repeated here:
Definition: Let $X$ be a scheme. A sheaf of $O_X$-modules $\mathcal{F}$ is called locally free of rank $r$, if there is an open cover $\{ U_i \}$ of $X$ such that $\mathcal{F} \mid_{U_i} \simeq O_{U_i}^{\oplus r}$ for all $i$.
Definition: A vector bundle of rank $r$ on a scheme $X$ over a field $k$ is a $k$-scheme $F$ and a $k$-morphism $\pi:F \rightarrow X$, together with the additional data consisting of an open covering $\{ U_i \}$ of $X$ and isomorphisms $\Psi_i: \pi^{-1}(U_i) \rightarrow U_i \times \mathbb{A}_k^r$ over $U_i$, such that the automorphism $\Psi_i \circ \Psi_j^{-1}$ of $(U_i \cap U_J) \times \mathbb{A}^r$ is linear in the coordinates of $\mathbb{A}^r$ for all $i$, $j$.
It is argued later, that free rank $r$ sheaves are in 1:1 correspondence with vector bundles of rank $r$. Is this to be understood as: Fix $X$ to be a scheme over $k$. Then vector bundles of rank $r$ on $X$ are in 1:1 correspondence with free sheaves of rank $r$ on $X$.
I am not sure, whether or not I understood correctly, because $k$ is mentioned in the second definition but not in the first.
Also, I do not understand what a $k$-scheme is. (Neither our lecture nor Hartshorne gives a definition, the only definition I found that sounds similar is definition 32.20.1. of the Stacks project. However trying to understand this a get lost backtracking dozens of previous definitions) But it seems that in this case it is enough to think of $F$ as a scheme over $k$. The other $k$-scheme properties will be forced on $F$ by the special local structure. Is that correct?