Your scaring me. Ok. First of all the ONLY thing you are going to change in the lamp wiring is not so much a change but the wire
labeled HOT needs to be cut (with the power off please) and a SSR solid state relay inserted with the output screws connected to the two ends of wire you cut. The arduino will turn the light on and off (if desired) with a digital output pin connected to the +V input screw (dc voltage input) and the ground for the input is connected to the arduino. The SSR will provide isolation from the 120vac. Everything else on the lamp stays exactly the same.