I think it might make sense to show a bit longer on the video if you can. It cuts out right at the interesting part for me. For example, it looks like it's still idling at 1k after the button press... but what happens if you move off, and stop again... does the engine actually stop?
Have you connected any kind of code reader to the car, see if there's any faults with that system?
-edit-
And as I wasn't sure I tested at lunch whether the start stop resets when I change between normal and track. It does not. It's totally independent in mine. I get in press track, and the stop start... or vice versa... then got to the end of a drive and pressed normal, and stop start was still off. Are people really seeing that turn on again when they change modes?