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.
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.
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
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 :-).
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.
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.
That's a good mathematical model of the behavior of the elements from a pedagogical perspective, though confusing for more advanced readers.
(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)