The braided line (that's cut) was your cat's temp sensor. You most likely have a de-cat test pipe that doesn't use the exhaust temp probe.
Your reverse lights are wired to the trans. and should be on the forward switch (if you have a manual) on the driver's side. It should wire into a connector on the driver's side of the engine bay.
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1990 Skyline GTE/S-t 4 Door Updated 6 Mar
Swapped out RB20E with RB20DET
Trans converted from Auto to 5-Speed...
Now with SR20DET 5-Speed
Installed Type M 5-lugs and brakes, R32 GTR rims and 2-Dr rear end (Tail lights & bumper)
1994 Isuzu AMU
US Spec Amigo with JDM Isuzu MU (Mysterious Utility, as it's called in Japan) nurf bars and hatchback style hardtop.