What type of wok apparatus is depicted in this home

The larger ring will transmit more energy, but spread over the size of the ring; so the density is probably roughly the same.

So using an element bigger than your pot/wok bottom is basically just a waste of heat/energy.

Some supporting evidence added by DMA57361:

I've recently gotten a free energy monitor, which tells me the current power consumption of my entire home. So, with a bit of experimentation, I have worked out the approximate power output of each ring on my hob, and this backs up that they have approximately the same output density.

Position     |  Diameter  |  Area   |  Power  |  Density
             |   cm       |   cm²   |   kW    |   kW/cm²
-------------+------------+---------+---------+-----------
Back-Left    |   18       |  254.5  |  1.90   |  7.46
Back-Right   |   16       |  201.0  |  1.45   |  7.19
Front-Left   |   14       |  154.0  |  1.25   |  8.11
Front-Right  |   18       |  254.5  |  1.84   |  7.24

I find it amusing that your range warns you the cooking ring will work as intended. It's supposed to be heat trap, and focus heat on the bottom of the wok.

That said, they are also correct that it may discolor the burner grate. I can't really say what your grates are made from, and many cooking materials discolor at high temperatures. I think the main idea behind this paragraph in your manual is to save them from warranty replacements for discolored grates.

I also find it highly suspect that a wok could damage the burner itself. They are usually ceramic, in direct contact with open flame, and any pan traps heat to some degree. Burners get HOT and have to be able to take it. Although the little decorative cap on the burner might discolor as well.