kmac

Its exactly the same as the circuit you built - instead of an AA battery there's a slightly bigger one. The main issue is with electronics in cars - you need something on ECUs and CAN BUS etc

May not be the sensor itself causing the errors - sometimes the metal ring gets rusted and broken causing ABS error codes. Happened on an old (1998 reg) Ford I have. Was a cheap fix, but fiddly - £2 for the reluctor ring (as they are called) from Fleabay (Ford wanted £40) and then just a case of removing hub and slipping the ring on. You can test each sensor with a multimeter - may need two people though. Jack up vehicle and put front on axel stands. Disconnect ABS sensor near the wheel and check resistance (ohms). Then spin the wheel by hand (this is where you may need a second person) and check that the ohms change. Indicates working sensor. Repeat for each wheel in turn. Fiddly, but google checking ABS sensors with mulitmeter for more info/videos on youtube.

The insulation also has an impact on AMP rating of wire - new thinwall cable can carry slightly more amps as it can dissipate heat better. The safest way is to get the wiring diagram for your vehicle and check what ratings the fuses are for the wires in question. Then replace with wire rated higher than the fuse. Wiring diagram can be found in the Defender Workshop Manual (some sites have this online but PM me if you need a link)