Will not start when the box on the park indicator hasn't turned red

For years my 1993 Integra 1.8 GS automatic transmission intermittently will not start. I believe it's due to a signal between the trans and ignition indicating that the vehicle is in gear, which will not allow it to start.

Other than not attempting to start (it turns over - battery strong - distributer and alternator are all good, and shift cable has been replaced), when the green shift indicator box surrounding the S3 flashes - the car will not start (this is a check I do when inserting the ignition key in the shifter lock release, and move the shifter to the S3/S4 positions) - if the box does not flash when in these positions, then I can move the shifter back to neutral or park, and the car will start).

It also will not start when the box on the park indicator hasn't turned red (fully in park). However, currently, in warmer weather here in Michigan, the car goes fully into park (red box around P) but still does not start. But again, if I perform the shifter check and the green box is flashing when placing the shifter at the S3/S4 positions, then when I go back to park or neutral to attempt to start, it will not.

Bottom line; when there's no flashing, the car will start, so the flashing has to be indicating something, correct?

Your thoughts?