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There is two or three wires coming off the reverse light switch on the steering column. The switch is further down the column (toward firewall), locate the green wire of the 2 or 3 that runs from the switch up the column. Then take your multimeter (a test light will work too), and set the parking brake and then shift the truck into reverse. When the truck is in reverse, the green wire should show ~12V+. If the fuse is good, and that wire shows 12V+ then your problem is after the switch (Bad ground, short, broken wire, etc).
If it was a bad ground between the switch and lights, the whole ass end of the truck would be going haywire. No lights at all. Trust me, it's happened to me twice.
Not all the time, a bad ground can cause all sorts of symptoms! From just dimming lights, to malfunctioning lights, to no lights, to only one working, etc etc. A bad ground can raise havoc with a lot of things. A broken wire is a very good posibility or the switch itself could very well be failing. I just outlined how to test it to see if it's putting out voltage or not @ the switch!
Jiggle the shifter a bit. On mine, the shifter has to be as far as it can go upwards (while still locked in reverse) for the reverse lights to come on. I can almost guarantee yours will do the same. Also, the pictures you posted of the side of the transmission is the speedometer gear. Not the reverse light switch. That is inside the steering column.