I assume that you have moved the torch connection at the welder from the positive of the rectifier to the inductor, and the work clamp and cable from the inductor to the positive of the rectifier?
If so, then the problem will be that the trigger switch in the torch has only a single thin wire, which connects to Pin 1 of the E1292.02 control board. The circuit for the operating coil of the relay ( which starts and stops the wirefeed and the welder arc ) completes by "borrowing" the heavy power cable in the welder torch as its second wire. This is joined back to the board by the wire on Pin 5, which attaches to the positive of the rectifier. However, that is now the work return cable, not the torch, so the relay cannot power on.
As Hutcho said, the "EN" Gas/Gasless versions of these welders have a 2-wire torch, with the second wire going to the positive of the rectifier, and Pin 5 of the board, regardless of the torch polarity.
This thread refers to polarity swap:
From the taped joints in your Pic 2, I think you've done the hard part of the polarity swap. Just need to get the relay to work....
www.mig-welding.co.uk
Option 1 - moving the wire at Pin 5 of the control board from the positive of the main rectifier to the negative - effectively following the movement of the MIG torch cable to the negative - will not work. The speed controller for the wirefeed motor is one function of the control board, and this needs power from the positive of the rectifier, so that wire has to remain connected. The other control board function - On/Off via the relay, the small transformer, and the trigger switch, could be enabled, but we would have to reverse-engineer the full circuit of the board and then cut a track somewhere.....
Option 2 - adding a second contact to the existing switch, or fitting a real microswitch inside the handle, is possible. I'm not sure how difficult it would be to insert a second thin wire inside the torch cable sleeve.
Options 3 and 4 are also possible, with perhaps 3 - a new 2-wire torch - being the simplest.