1. The popularity of the Saxonia Annual Calendar overshadowed the 1815 Annual Calendar. Many more people gravitated towards the former (Sax-0-Mat movement, iconic "Big Date" display, etc.), which was introduced first, even though the Saxonia AC lacks the one-push corrector for all calendar functions. That pushbutton on the Saxonia AC only changes the date -- nothing else.
2. I've never heard of anyone accidentally pushing a rectangular pushbutton on any Lange watch. I suppose it's possible, but I'd assume it wouldn't happen more than once every few decades.
Good luck in the hunt - keep us posted!