Eagle eyes lol on that relay - but yeah it’s facing the wrong wayThat is a great job of fitting the new transformer, very neat! I am really pleased that the welder works now, it always feels good to keep going, and not give up, and fix something. I was lucky to get that effect nearly every day, for the 44 years that I was a computer engineer. Well, sometimes it took more than a day... in one case, three months.
The power supply from a Dell computer is probably an ATX? If so, it does have a strong +12v, but the -12v ( which you could combine to get the needed 24v ) is most likely only rated at 1 Amp, which would not be enough.
Those front-panel pushbutton switches are a weak point on the SIP. I suppose they could be replaced with two good-quality rotary switches - a 1-pole 2-way for "1/2", and a 1-pole 3-way for "Min/Med/Max" ( or a 6-way multi-segment switch with some very fancy wiring ).
The MOV ( metal-oxide varistor ) is also known as a VDR - voltage-dependent resistor. It is used to protect against high-voltage spikes., such as can be generated by an inductive load - like a motor. As the voltage rises, the resistance drops, shorting-out the spike before it can damage sensitive electronics - such as the PWM motor speed controller. Diodes and a capacitor can be added for the same purpose, but I do tend to trust that these fiendishly clever Chinese PWM controllers can look after themselves.
The laptop power supply is best left powered-up whenever the welder is on, and the 20-ish volt output of that goes straight to the PWM controller. Then the two motor wires are detached from the original PCB ( retained inside the welder cabinet ) and connected to the new relay, as described in post #23.
Talking of relays, in the very clear picture of your relay on the PCB, photo number 3 of post #71, it looks to me that the rather burnt contact surface of the fixed outer contact leaf is facing the rivet head of the contact on the flexible moving inner leaf, rather than the large domed gold-plated contact surface. In other words, that relay was assembled with the moving inner leaf back-to-front, which would explain why the contacts burnt so easily.