Diff year and gen but here goes. On my '97 I had this exact same issue - key is out, but opening the door you got 'key in ignition' (but not if key was 'on'). On mine, if you take off the clamshells on the steering you can see the little contact, in this case it wasn't secure in its position and could make contact where it wasn't supposed to. I took a small strip of tape and wrapped around there to hold it, problem solved, it works properly. Admittedly that's a tape job not a "real fix" but it solved it, and, if it comes back, which it hasn't, I'd know right where to look. Not sure what the "real fix" is but it looked like it might involve replacing the ignition cylinder which I'd rather stay away from if possible.
Anyway if yours is at all like mine and you take off the clamshells you should see the contact for this function and probably it's floating around in there. Then you can move it around and probably make the chime go off and that tells you where it's supposed to be, and secure it there somehow.
Maybe could be other causes too but I thought I'd throw that out there as it sounded real familiar.