Do I need thermocouple wire? But even if I used that wire, there would be a copper/solder/wire junction at the PCB. What care do I have to take there? Is it acceptable to solder copper wire to the PCB and connect it to the plug with the screw terminals? Is is also still acceptable to solder screw terminals on the PCB and connect the wire there at that end?
If you are connecting your TC directly to the module you are OK, the module has cold junction compensation built in, but if you want the TC farther away than the length of the TC wires you must use TC extension wire. You only need copper wire from module to Arduino.
You have to know the temperature of the junction, where different materials cause a voltage difference. The cold junction compensation on the module takes into account the temperature in exactly this place. Shifting that "cold" junction to some arbitrary place, of an unknown temperature, will result in false readings.
TC extension wires cause the same offset voltage on both lines, not adding any errors. They differ from the TC materials in price, maximal temperature and other practical factors.