I guess the "high-altitude" reason is that $L$, the group of principal units of the discrete valuation ring $\mathbf{F}_q[[x]]$, is an abelian pro-$p$ group. If $A$ is any abelian pro-$p$ group, then there is a unique structure of $\mathbf{Z}_p$-module on $A$ which is continuous in the sense that the map $\mathbf{Z}_p\times A\rightarrow A$ is continuous and which extends the given $\mathbf{Z}$-module structure $(n,a)\mapsto a^n$. It is defined as in your first definition: take $f\in L$ and $z\in\mathbf{Z}_p$, and write $z=\lim_nz_n$ for integers $z_n$. Then the sequence $f^{z_n}$ converges, and we define $f^z$ to be the limit. This turns out to be independent of the choice of sequence converging to $z$.
Regarding Witt vectors, the ring $\mathbf{F}_q[[t]]$ is an equicharacteristic $p$ discrete valuation ring, whereas $p$-typical Witt vectors over perfect fields of characteristic $p$ are mixed characteristic discrete valuation rings. Namely, $W(\mathbf{F}_q)$ is a complete discrete valuation ring of characteristic $0$ with uniformizer $p$ and residue field $\mathbf{F}_q$. Again $1+pW(\mathbf{F}_q)$ is an abelian pro-$p$ group, and so again it is canonically a $\mathbf{Z}_p$-module.
Now, the binomial series definition makes sense in characteristic zero, because $\mathbf{Z}_p\subseteq W(\mathbf{F}_q)$, and one can make sense of the $\tbinom{z}{k}$ for $z\in\mathbf{Z}_p$ and $k\in\mathbf{Z}_{\geq 0}$, but I'm not sure if this continues to work in characteristic $p$. If it does, then it can be used to define $\mathbf{Z}_p$-exponentiation in $A$, and then, if you wanted to prove that this definition agreed with the one described above (your first description), you could show that both extended the usual $\mathbf{Z}$-exponentiation and that both gave $A$ the structure of continuous $\mathbf{Z}_p$-module. Then uniqueness (really density of $\mathbf{Z}$ in $\mathbf{Z}_p$) would imply that they coincide.