I am writing this to relay how I resolved my problem.
Opening the LH drivers door, the LH under dash, both door and both back seat pillar lights came on. Opening the RH passenger door, only the RH under dash light came on. I tore things apart and traced wiring trying to find the problem.
What I finally found was very interesting and weird. The wire numbers I will use refer to the 3-E9 figure for 1968 Cougar Interior Lights. I also have a Group 19 schematics page 19-44, Figure 31 that shows the same information.
As found, the RH door lamp wire 53 had been unplugged from the Blue single pin connector. This wire was connected to a wire coming from the LH side of the car that became hot when the LH door was opened. The female end Blue single pin connector shows wires 53F and 53E spliced together. When the RH door opens, this should turn ON the LH side lights. I probed the Blue connector with a volt meter and found 12 volts when the RH door was open. I then concluded that there had to be a discontinuity in wire 53E or the splice at another connector with wire 53D. I couldn’t find the blue 53E/53D splicing connector so decided to try to put things back like the wiring diagram with the RH door wire 53 plugged in to the blue connector. I shoved a male pin into the Blue female connector and BEHOLD, the LH and back seat lights came ON. Somehow, in that single pin female connector, the 53F and 53E wires weren’t actually making contact. This is the interesting, weird condition I found. Somehow, those two supposedly spliced wires weren’t making contact inside the Blue connector.
I hope this tidbit of information helps someone else with a wiring problem. I think that the wires should be connected whether the male is in the connector or not, but in this case it was very misleading.
I connected wire 53 as the schematic shows and everything works right. The wire from the LH side that had powered wire 53 I taped off.
Thanks to anyone who responded to my original post titled 68 electrical problem.