the forum is definitely the right place to ask for support, any other channel will not get Gus’s direct input 
Thermocouple module https://www.ghielectronics.com/catalog/product/397 ? By two connectors do you simply mean the double-ended spade lugs that the image on that page shows as being on the non-sensing end of the thermocouple? Those lugs are not designed for a simple screw-down connector so you can safely use them with one arm of the U shaped lug in the screwdown connector, and I suggest you leave the other end protruding from the connector on the outside of the board so there’s no chance the two will touch at all.
As for internal and external temperature, the definition of those is based around the actual chip used on the module for sensing. Thermocouples are sensitive devices and the amplifier chip adjusts the temperature read from the thermocouple based on what the temperature is at the"cold junction", i.e. the temperature of the chip, in what is called cold junction compensation. It would appear to me that the internal temperature is a measure of the CJ temp, and the external temp will be the compensated measured temperature. If you want to dig deeper here, the TC chip in use, MAX31855, is documented Mixed-signal and digital signal processing ICs | Analog Devices