Option three in that FAQ is not what you are describing. The switch on the wall does not control the current to the lights.
Instead, that switch allows for either a tap or a double tap. The tap cuts the current on the circuit branch the switch is installed on. The double tap does not change the current directly. Instead, it sends a radio signal either to the smartthings hub or to the bulbs directly if they are Z wave smart bulbs (so not to, for example, Hue, which are not Z wave)
The single tap works like a regular dumb switch and turns the current on and off. In the use case described in the FAQ, that would turn off the fan, but not the lights. The lights would normally be turned off with the pull chain.
However, if you use smart bulbs, then you can leave the pullchain current always turned on and communicate with those bulbs by radio.
That allows you to use the double tap feature (which again, does not cut the current) To turn the bulbs on and off independent of the pull chain.
I hope that helps clear up any confusion.
If you have your switch on the wall, whether it is smart or dumb, wired in such a way that just turning the switch off turns off both the fan and the light kit, that’s very unusual and it’s not how most fans are designed.