NO Don't CUT in two, just make carve away the plastic that cover the wire, carve about 1/8 of an inch.
about | | << see that two line? carve away that much. that's all you need. Don't cut the red wire into two piece. After you carve the lenght above, take a toothpick and separate the small wires in between the place you carve, insert the wire from the neon in between the red wire and wrap the neon wire all around 1 or 2 round. Tape it with electrical tape and you're done. Before you tape it up, test it first ok? if the neon light up then u're doing right, if it doesn't light up then it's the other red wire with 1 color red wire. So try that. But i think the red wire with spot is the right one. Look at the attached picture.