Garage light switches don't always work
After 10-12 years, it's entirely possible that the switches behind the light buttons are failing. This would be especially true if you use the garage door opener lights on a regular basis. Consider that you probably hit the opener button (the big white one) once in the morning to open the door on your way out (then close it with the remote in your car), then once in the evening when you return home (after opening the door with the remote in the car). If you go to the garage twice in the evening, that's 4 presses of the light button (on/off for each trip), therefore more wear &amp; tear on that particular switch.
If you're good with soldering, you could attempt to open up the wired remote, desolder the actual switch from the circuit board and replace it.
However, it's probably easier (and probably wouldn't cost significantly more) for most folk to simply replace the wired remote. You could get a Chamberlain, LiftMaster or Craftsman brand to replace it (since they're all made by Chamberlain), generally, though, these are pretty generic and almost any one will do.
I'd suggest replacing one, then testing for a few days to confirm that the light button works consistently. If that's solved the problem, then replace the other.
Replacing the wall-mount remote should be as simple as:

Pop the white plastic cover off
TAKE A PICTURE NOW
Disconnect red &amp; white wires that are attached
No, really, you did take a picture with your phone back there, didn't you?
Unscrew 1 or 2 screws that hold the unit to the wall, and pull it off the wires.
Feed the wires through the hole in the back of the new unit
Screw it to the wall (existing screw holes if they happen to line up, otherwise, it's just drywall, screw it in wherever the new mounting holes are)
Attach the wires to the new opener (most likely exactly the way they were attached to the old one, but check the directions, just to be sure).
Pop the new white button onto the newly mounted remote.
Test!
No, really, you took a picture before you disconnected the wires back there, just to be sure, right?