"I think Moses can do a better job, but a stab -- your example is like a
parallel plates cap. connected by a wire. Here the current and current
density are identical, except for pinch, skin, size, etc. effects. So
I'm stuck -- what's the problem?"
I, personally, do not see a problem here either, although I do not agree that the current and current density are identical, even in this example. At the risk of getting annoying by repeating myself, current is an integral characteristic of the process of charge transfer and a scalar, whereas the current density is a local characteristic and a vector. I also want to add what I should have mentioned from the very beginning, that in cases when we do have a certain direction singled out, (like in case at hand), we use what is called the CURRENT ELEMENT: dI = Idl - that is the product of current I and the incremental displacement vector dl along the direction of propagation. This is suited specifically to treat the problems mentioned in this discussion and also when we have thin current filaments (see, e.g., J.D. Jackson, Classical Electrodynamics, 3-d ed., John Wiley & Sons, p. 175 and further on.), and shows again that current itself is a scalar quantity. A moving point charge qv mentioned by John Denker, while being a vector, is NOT a current, but yet another example of a current element.