Ok, we're having a terminology situation here.
I think everyone has gotten that the DRLs are the "halo" of LEDs around the primary headlights.
Now we get to the meat of the issue... In Mini parlance, "cornering lights" are an option that physically move the primary headlighs as you turn the steering wheel into the corner... and they are an expensive option... in the US, a $1,000 option, in fact. The small, circular lights in the front fascia that are auxiliary to the main headlights are divided in half and have two functions. Under normal night driving conditions with the LED headlights on, one half of the little fog lights automatically illuminate with the primary headlights (and the DRL halo as well). They are just additional illumination, but very effective in lighting the road ahead of you, as they are focused lower than the projected beam of the LED primary headlights. Basically, lighting up the shadows caused by the projector beams. When you press the fog light button, the second half of those little lights in the bumper fire up, casting a much brighter, and much wider beam underneath the main LED headlights.
As for standard/option (and I just popped onto the configurator to confirm this is still the case): Halogen headlights and halogen foglights are still the standard configuration on the Cooper S/F56 in the US. LED headlights are a $750 option, but the LED fog lights come with them at no extra charge as part of the package. Interestingly, now when you roll your mouse over the LED headlight option, a little message pops up that says it's an "included item" and as you mouse over the LED fogs, it says, "required item". So it looks like they at least took a step toward unscrewing some of the confusion over the $0 fog light option.
Hope that helps