Some meters always display the cold junction temperature, whether a K-type thermocouple is connected or not. It is documented and is a feature.
In all meters, the measurement is taken close to the reference junction (ambient), with a thermistor, RTD, thermal diode or temp sensor IC.
Once the thermocouple is plugged in, that temperature is used for cold junction compensation.
The hot junction temperature can then be calculated, using voltage and look-up tables (or formulae) to compensate for non-linearity.
Shorting the probes, mimics a thermocouple with no induced voltage and the meter only displays the remaining cold junction temperature with its compensation (ambient).
Really?
My Fluke 287 show me 26,2C with short circuit leads and 24,7C with connected thermocouple to the end of this leads. So, what is reference and for what this reference, when type K thermocouple junction voltage well know and constant more or less for all thermocouple probes?
What reference temperature (in mV for processor) in digital thermometers that supported more than one types external thermocouples?
Have you even tried connected to your meter Type K thermocouples, already being some time under high or low temperatures to compare with DMM environment temperature?
IMHO, internal temperature that DDM show you with short circuit leads only for show user that temperature coefficient must be applied to the readings, when internal DDM temperature stabilized away from calibrated diapason 23+/-5C. It easy can be checked with just short circuit leads, no matter have you external thermocouple or not.
When you have external thermocouple, especially with fast response, internal and external temperature always will different with thermocouple connected and internal DDMs temperature can't be reference.