You seem to be missing the fact that current density J is for a continuously-flowing current moving through some unit of area. In your example of a single char

You seem to be missing the fact that current density J is for a continuously-flowing current moving through some unit of area. In your example of a single charge moving at velocity v, which passes through the origin of your coordinate system at time t, the current density at the origin is basically a Dirac delta function. In other words, there is zero current density through that point at all times other than at one instant (vt) where it would be infinite (although finite when integrated over time, basically how Dirac delta functions work). Thinking about it in this way may help you unite the gap between a moving and stationary charge. You can also think of a Dirac delta function as a finite but normalised Gaussian (or normal) distribution of charge, which gets rid of the discontinuities. When that makes intuitive sense, take the limit where the width of the distribution approaches zero (but remains normalised so the amplitude becomes infinite).