The classical definition of subnet seems to be that $\Psi: J\to X$ is a subnet of $\Phi: I\to X$ if there exists a monotone, final map $h: J\to I$ s.t. $\Psi = \Phi\circ h$. I found another definition that states that $\Psi$ is a subnet of $\Phi$ if for every subset $A\subset X$ holds that $\Psi$ is eventually in $A$ whenever $\Phi$ is eventually in $A$. Here $\Phi$ eventually in $A$ means that there exists $i\in I$ s.t. $\Phi(k)\in A$ for $k\geq i$.
The second definition has the advantage that it is better suited to translate between nets and filters.
My question is: beside the point with the filters, are there differences between those two definitions with "real" impact?
Probably not. This PDF of notes by Martin Sleziak compares the three common definitions of subnet; you’ve given the definitions of what are there called AA-subnets, the most general type, and Willard subnets, the least general. The final result is that if $\mu=\langle y_\beta:\beta\in\Bbb B\rangle$ is an AA-subnet of a net $\nu=\langle x_\alpha:\alpha\in\Bbb A\rangle$ in a set $X$, then $\mu$ is AA-equivalent to a Willard subnet of $\nu$, meaning that $\nu$ has a Willard subnet that generates the same eventuality filter on $X$ as $\mu$. (The eventuality filter of $\mu$ is just the family of subsets of $X$ that contain a tail set of $\mu$.)
Thus, the choice of type of subnet is should be just a matter of convenience for just about any result that can be stated in terms of filters.