HP Tuners calls these check boxes "SES Enable". TunerCat OBD-2 Tuner refers to them as "DTC Enable" (which sounds like the same thing DHP refers to them as).
The problem is I don't think anybody has the definition for these check boxes entirely correct. In factory stock cal's I have seen these boxes NOT checked for trouble codes that do turn the check engine light on. I do know in the 05 Monte SS cal I tuned, even when setting the DTC to "No MIL"; leaving this box checked resulted in the Driver Message Center lighting up with a warning message stating the vehicle needed to be serviced as soon as the code set (after the cat diagnostic had run). I had to uncheck these boxes next to these CAT and post-CAT O2 sensors codes to prevent the Driver Message Center from displaying these warnings.