You want to implement the symmetries as transformations on the fields (and also corresponding operators on the Hilbert-space in the QFT), i.e. transformations of the dynamical variables in your Lagrangian $\mathcal{L}(\psi, \bar{\psi},...)$. The reason is that these symmetries of the Lagrangian have important consequences: Continuous symmetries for example give classically conserved current through Noethers theorem and Ward identities in the quantum theory. Discrete symmetries, like $C,P,T$ are also important.
Now let us apply the charge conjugation, which you defined above, to the QED Lagrangian. One can see that
$\bar{\psi} \partial_{\mu} \gamma^{\mu} \psi$ and $\bar{\psi} \psi$ are invariant, but $\bar{\psi} \gamma^{\mu} \psi \rightarrow - \bar{\psi} \gamma^{\mu} \psi$, so the QED interaction term transforms as
$$
e A_{\mu} \bar{\psi} \gamma^{\mu} \psi \rightarrow - e A_{\mu} \bar{\psi} \gamma^{\mu} \psi,
$$
which is the same as taking $e \rightarrow -e$. This is one easy way of seeing why charge conjugation is an appropriate name. Now if we let $A_{\mu} \rightarrow - A_{\mu}$ under $C$ we get a symmetry of the QED Lagrangian and a consequence is for example Furry's theorem.
I also strongly recommend you to read p.192-195 in Schwartz's QFT book, where the explanation is more detailed and there is a more correct justification of the name charge conjugation.
