Min Power is used for cuts where the head will be changing directions, slowing down the speeding up again, where you want to not over burn the corners. DSP controllers like the Ruida will ramp, within the range of the Min to Max Power based on the speed the head is traveling. Min Power set too high, over burned corners. Set too low, the laser will not “fire” and will not mark/cut in the corners. The firing point is the ionization point of the glass tube. The bigger the tube, the higher this point, which is why some “heavy cutting” machines include a 2nd, smaller tube for engraving.
LightBurn also uses the Min and Max range when using the ‘Power Scale’ feature.
The min power setting is ignored for everything but grayscale on GCode based devices. The setting for this is in firmware, and can’t be specified per-cut like the high-end DSP controllers do.