This is not even approximately true. j is not “defined in terms of E”. It occurs as a source term in one of the inhomogeneous Maxwell equations, and consistency with the other inhomogeneous one (div D=rho) requires that div j = dot rho, which means that j is the flow of charge density. So, I guess if electron movement shouldn’t be referred to as current, we shouldn’t refer to electrons as charges, either…
When solving for the wave equation, the substitution made for J is sigma * E. So defined in terms of the E field and some conductivity constant. When solving the wave equation, there is no mention of charge density, everything is fields. This predicted the existence of electromagnetic waves. In reality, we know there is only electrons and photons. So thus the wave equation is the source of truth and what we call fields are just a consequence of the complex interactions between electrons and photons and this is studied in quantum electrodynamics.
