As I suggested, examine the physical point the shifter lever is attached to the shaft, and verify the punch marks line up, as in the picture below, this part is easy, and external...
then, verify the "angle" of the lever mechanism is at 90*, noted by the point B in the second photo...
Finally, if this has been verified, the next step is to remove the clutch cover, and the complete clutch pack, to access the "external" shifting "star/actuator/fork" mounted on the end of the shift shaft... as seen in the third picture...
this has been known to have a couple issues in the past. Broken springs, bolt backed out, locating dowel pin behind sheared, and loosened "stud" that holds spring... any of which could be the cause of an "incomplete" shift, or "false neutral"...
OK... edit to add;
after looking closely at the other thing you said...
Display shows properly. 1st. 2nd. Nothing there 3rd.
I'm asking you to clarify a bit... first and third show up on display? and second does not? or am I missing something...
I ask, as the electrical contact to send the signal to the gear "location" switch, is actually made by 2 pins, located on the end of the actual shifting drum... so a lack of "complete" rotation on the drum, to the "next" gear, as would occur from what I noted about the star mechanism and associated fork/lever, would in fact "short shift" and the sensor would show nothing in second gear.. also... what happens is in the "short shift", the drive gear does not fully engage second gear, stops short, then when shifting to third it
will go into third.. I say look again at the lever, then follow up with the mechanism behind the clutch basket that rotates the physical shift drum.
hope that helps a bit... it's in section 9-37 of the FSM for your model year.