Why define $D_\mu = \partial_\mu -ieA_\mu$ with the electric charge $e$? If $D_\mu = \partial_\mu - ieA_\mu$ then the QED Lagrangian is invariant under
$$A_\mu \to A_\mu + \frac{1}{e}\partial_\mu\alpha(x)$$
$$\psi \to e^{i\alpha(x)}\psi$$
However if $D_\mu = \partial_\mu -iA_\mu$, the transformation needed for $A_\mu$ is simpler:
$$A_\mu \to A_\mu + \partial_\mu\alpha(x)$$
The lagrangian is still left invariant by this transformation. 
What is the reason to add the electric charge to  the definition of the gauge covariant derivative?
 A: The answers in the comments are gone and so in case someone else has the same question I will type it here but not accept the answer as it is mostly not my own.
Benefits of defining $D_\mu = \partial_\mu - i e A_\mu$ include:


*

*The coupling constant between electron and photon $e$ is written explicitly

*The conserved current becomes $j^\mu = e \bar{\psi}\gamma^\mu\psi$, which gives an electric charge operator proportional to $e * (Number\,\, Operator)$
The conventions with and without an explicit $e$ are both used: Without the $e$ in more abstract settings which do not require explicit calculation, and with the $e$ in (for example) phenomenology. 
A: In your QED Lagrangian you have 
$$
\mathcal{L} = \bar{\psi} ( i \gamma^{\mu} D_{\mu} - m ) \psi  - \tfrac{1}{4} F_{\mu\nu} F^{\mu\nu}
$$
where $F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$. The field $A_{\mu}$ are your photons, and the field $\psi$ are your electrons/positrons.
$\mathcal{L}$ tells you the rules for calculating things in QED. Focus on the following term in $\mathcal{L}$
$$
\bar{\psi} i \gamma^{\mu} D_{\mu} \psi = \bar{\psi} i \gamma^{\mu} ( \partial_{\mu} - i e A_{\mu})\psi = \bar{\psi} i \gamma^{\mu} \partial_{\mu} \psi  + e\; \bar{\psi} \gamma^{\mu} A_{\mu} \psi 
$$
You see that there is now a combination of fields $e \bar{\psi} \gamma^{\mu} A_{\mu} \psi $ appearing. This describes how the photons and electrons interact with each other $\to$ so a way of thinking about why $e$ is there, is that it is a measure of how strong these particles interact with each other. 
