Whoever informed you the shaft should move may have misguided you. The part that moves is internal. I am including a pic of the layout. Part #6 is the pin that moves. You wont be able to see that unless you unscrew the shaft. I would try and put the new coil on the old shaft and see if you hear the pin inside 'click'.
(P.S., I have worked at a John Deere dealership since 1998 as a lawn and garden tech/shop manager. Just to give you a bit of background on my experience level.)
Good luck and let me know how you make out!
View attachment 44560