This chart should give you a rough idea
This is the same chart for ALL K type thermocouples regardless of who makes them or the length of the probe.
Whats different between a 400C rated probe and a 600C rated probe could be the outer metal of the probe. The general construction of the TC.
Doing the ice water test to verify 32 degrees F is fine but the problem is that isn't near the standard operating range that you would be using it at.
So imagine tuning a car to run perfectly at idle compared to a 4K RPM.
If the thermocouple is designed exactly to the spec then you are perfectly fine calibrating your PID at 32. But I doubt that a $5 Thermocouple will be built exactly to the K spec.
it will be close though..
We are casting boolits. It doesn't matter if we are off a few degrees.
What you want with a PID is repeat-ability.
Basicaly knowing that at a certain temp (lets say 700 F) that every time the PID reads 700 its gonna be the exact same temp as it was before.
The reason I say that temp doesn't really matter because if you aren't casting good boolits at what it says it 700 then you turn it up or down.
It doesn't matter if the PID is off a couple degrees or 50