IMO 'covariant formulation' of electrodynamics means that the equations should remain invariant across different Lorentz frames. Now there are broadly two ways to write electrodynamics equations.
using 3 vector notation $$ \frac{\partial \rho}{\partial t} + \nabla.\mathbf{J} = 0 \tag{Eq. 11.127 of J.D.Jackson} $$
using 4 vector notation and tensors $$ \partial_\alpha J^\alpha = 0 \tag{Eq. 11.129 of J.D.Jackson} $$
An electrodynamics equation written in either of these two ways remains invariant (form-wise) across all Lorentz frames. Then why is it that the latter is referred to as the 'covariant formulation' but not the former?