If I have a piece of wire with a constant current through it the charge everywhere on the wire remains constant. In fact the charge will be more or less neutral in most cases.
Note that the alternative would be that the wire somehow accumulates charge as it passes current, which clearly isn't happening.
No, you are wrong. Electrical current is the mathematical representation of a charged particle moving between two points. The value of the current is directly related to the amount of charge being moved.
As for your "wire" example, you're conveniently ignoring that a single (open-ended) wire isn't a closed circuit and as such there can be no current (and hence no accumulating charge). And if it's a closed circuit, then there are other elements in the circuit that consume or generate charge.
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.
Fine, charge a circular piece of wire, place it in space and spin it around. Voila, current without any accumulation of charge.
Honestly I'm confused why it's turning out to be such a controversial point that there's a difference between a change in charge and the movement of charges.
Heck in their diagram they claim that the voltage across an inductor is the double derivative of its charge, but inductors can't even hold charge so it's unclear what they're trying to say. They also claim that the charge across a resistor is somehow related to the integral of the voltage across it, but again (ideal) resistors can't really hold charge. The only way to interpret this supposedly 'universal' periodic system is by interpreting charge and its derivatives in different ways depending on context, which isn't convincing in a supposedly universal system.
It's ok, if I didn't have a Master's in the field I would probably have similarly down voted you.
The problem is mainly that the criticism you are making is not great for pedagogy. What is being called “charge” is probably something like “disposition to accumulate charge” or so, in the same way that force is not actually mass times acceleration, but it's mass times a disposition to accelerate, so that you can do things like measure my weight-force even though I’m not falling through the floor.
The dispositional truth of the matter is fundamentally more cognitively complex to teach than the simple rule that you get when you say that everything does what it's disposed to do, and so everybody has memorized the version of the definitions that has no dispositions, and gets very confused when you point out that aspect of those definitions.
I suppose I also didn't provide an awful lot of explanation to my point, but I figured I could just explain when asked. I didn't expect the difference between current and a change in charge to be this controversial.
Except there has been no controversy at all. Everybody understands that current (i. e. flow of electric charge, water, etc.) may have nothing whatsoever to do with "change in charge" (or in the mass of water) contained in a volume of space through which charge or water flows.
If what's being called "charge" here is charge distribution between one side of the element and the other side within the circuit (basically, electrons to the left minus electrons to the right, divided by two lest we count a single moving electron twice), and what's being called "current" here is simply current across the element, then in this case, i = dq/dt.
That's a good mathematical model of the behavior of the elements from a pedagogical perspective, though confusing for more advanced readers.
I believe there is another case, very similar (also involving a closed loop): If you pass a (properly oriented) magnet through or even across the opening of such a loop, an electrical current will flow around the loop (while the rest of the atomic bits will stay more or less in the same place ;) ) and no charge will accumulate.
(This caught my attention because I had up to that point believed that electrical fields are always conservative, and this demonstrates that not to be true: an electron traveling around the loop in such a manner will eventually return to its starting point having done non-zero work)
They are both correct: I = dq/dt is true only for scalar flux - a fixed, flat, surface with constant charge applied perpendicularly at every point across it. A popular simplification/special case of the general equation from vector calculus.
Note that the alternative would be that the wire somehow accumulates charge as it passes current, which clearly isn't happening.