You are correct in your assesment that the switch on the transmission is the culprit. It is the one on the right side with a single wire coming out of it.

Your pushing the shift lever to the right which then caused the OD to engage was the clue.

At this point it does not matter what any manual names the switch. At the dealer, ask for the 4th gear switch for the OD. When they give it to you, it should have only one electrical connector.

One trick to make the switch easier to access.

After supporting the transmission with a floor jack , the type on wheels with a cup on the lifting arm, remove the left side transmission support bolts one at a time then reinstall them three full turns. Then remove the right side support bolts all the way. By slowly lowering the transmission, the tranny will roll slightly to the right as the support hits the two left side bolts, exposing the switch. Lower just enough to access the switch.

When re-aligning, use a guide drift or long Phillips screwdriver in one of the right side bolt holes to re-align the support as you jack it back up.

Duane