Sounds like the contact is broke, bent or unable to make a connection when in the 'locked' position. Could be a simple fix but you'll need to look in there to see. (not fun)
For the transmission, usually the arm type shift linkage is on the 'i' trans. and the plate style is on the 'e'. But my car has an 'e' transmission and I have installed the 'i' arm style linkage. if you remove the shift boot under the black rubber cover thing you'll be able to see if it's the plate or the arm style linkage.
Performance one step at a time.