[drostie]

They are not wrong. In the Maxwell equations both come in as fundamentally different terms.

The claim is that current density J is different from the time rate of change of charge density dρ/dt.

That is not to say they are unrelated; they are related by the continuity equation,

    dρ/dt = -∇·J.

The distinction is real, because what you are calling current in the one case is actually a spatial derivative of current, as indicated by the ∇.

I would actually go a step further than this and say that current is actually properly defined as the source of magnetic field. On the conventional definition of current, it is physically impossible for current to flow through a capacitor, but we speak of that all the time. So the True Current Density is just

    J + ε dE/dt

in SI units. Actually taking that seriously, however, does require to committing to language which sometimes seems a little awkward, like saying electromagnetic radiation involves an AC current oscillation that propagates through empty space transverse to its oscillation.

[alxlaz]

> I would actually go a step further than this and say that current is actually properly defined as the source of magnetic field.

That's actually (still, and somewhat) how the Ampere is defined. There are ongoing efforts to change this though.

I... prefer to avoid discussions like this one, but I thought you might appreciate this part :-).