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?