Reverse light

Are you talking about the 'Reverse' light under the dash or your back up lights? Either way, your linkage should not have anything to do with it. If your transmission goes fully and properly into reverse, the reverse light switch on your tranny should send power to the 'Reverse' light under the dash and your back up lights. The reverse light switch should be a 2 prong switch. One of the wires is switched power going into the reverse light switch, and the other is power going out to the 'Reverse' light under the dash and the backup lights when the tranny is in reverse. Try this test with a circuit tester. With the ignition switch ‘ON’ (and the car not running), one of the wires going to reverse light switch should be hot. If not, look for a fuse or some other problem like a bad connection. More than likely, you will have power going to the switch. Then put the transmission in reverse. If the switch is working properly, the other wire will now be hot. If not, the switch is likely bad. I guess there could be something wrong inside the transmission, but most likely the problem will be the switch. Let me know how the test works.