That's the thing though, in the article the charged object does not move relative to the rod. The rod, object, and person B are stationary relative to each other. I was actually really surprised to read that two charged objects moving with the same velocity relative to each other generates a magnetic field from A's perspective.
Two long charged rods moving with the same velocity relative to each other are two parallel wires carrying current, the second of which is a typical and easy-to-calculate example of electromagnetic attraction. In fact, that situation is so prototypical that it is used to define the Ampere in terms of what current is required to produce a certain force between parallel wires. [0]
In this scenario, rods are stationary but they have current running in them - a different situation. Can it be translated to the relativistic one?
Maybe what they really wanted to say is "there is a speed c at which the charge of a moving rod is indistinguishable of current running through stationary rod"? Now that would make a lot of sense.
It's not different as far as electrodynamics is concerned. The motion of the charge is what matters; whether or not the neutral part is stationary or moving is of no concern.
