To which part of the quote do you refer?
"Modern electronic Machmeters use information from an air data
computer system to correct for temperature errors. These
systems display true Mach number." is not strictly true as there are more errors than temperature.
Checking my notes (prehistoric), and in very simple terms, Mach number (as measured by a machmeter) essentially compensates for temperature/density changes as they sit on both sides of the 'diaphragm' between the capsules ie if the air is warmer (less dense) then TAS needs to be higher to create the same dynamic pressure, which 'compensates' for the higher speed of sound in that air (although I have always been suspicious of the accuracy of this in a sort of luddite way). I believe the equation for Mach No does not in fact include temperature, but that bit of learning has long since vanished.
For ATCast - NB you refer to 'TAS' but remember the machmeter is using IAS, hence the 'automatic' compensation?
I therefore vote the whole quote as misleading, at best.
EDIT after digging around in boxes....