Frankly though, I not sure why they couldn't just use a simpler correction: postulate that moving charge only creates magnetic field with regard to objects moving relative to it. Which is also basically true?
This way, the stationary observer will sure notice that there's F_B, but will also know that the charge moving alongside the (also moving) rod does not experience it.
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.
This way, the stationary observer will sure notice that there's F_B, but will also know that the charge moving alongside the (also moving) rod does not experience it.