Electronic – Where is the cold junction on the commercial thermocouples

Wire ends can be twisted together without any problems.

I performed an experiment as part of my undergraduate coursework where we used twisted pair of copper and constantan wires (twisted part about 2cm in length) that were held in place mechanically. It worked quite well. The temperature range was about 100-300C, so that should be right in your ballpark.

When you twist the wires, you have to make sure they are twisted around each other, not one twisted around the other. The best way I found to achieve this is to bend each wire 45 degrees, so they make a 90 degree angle with respect to each other. Then just hold both wires and twist.

... what sort of accuracy expectations demand cold-junction compensation as opposed to simple calibration?

An ideal thermocouple provides \$\mathrm{V_{out} = k \cdot (T_{hot}-T_{cold})}\$.

The end result cannot be more accurate than the accuracy to which you measure or infer the cold junction temperature, and in practice will be somewhat worse and can of course be much worse.

Therefore, ALL thermocouple temperature measurements require cold junction "compensation".
"Compensation" is really shorthand for "determining the temperature of the cold junction with a level of accuracy that is appropriate for the application". Compensation could consist of saying "this will be used in an air conditioned office which is temperature regulated to 20 degrees C - so that will be the cold junction temperature".