The term "single threshold" is surely made up by the instrument supplier. I can only guess what it means, and I assume it means that the Cq is determined as the intersection if the threshold with the line connecting the two points of the amplification curve closest below and above the threshold. The regression analysis uses more than two points, usually the 3-5 points just above the threshold. The Cp is then the intersection of the regression line through these points with the threshold.

The "single threshold" method is better if the curves do not exhibit a clear log-linear phase (with 3-5 points in that phase) (however, in that case any quantification is "difficult" and not robust). When the curves do have a clear log-linear phase (as it should be), then the regression method is better, because it uses more data values and thus is less sensitive to noise (which is more effectively averaged out).

So the first thing is to use an assay that gives nice amplification curves with a clear log-linear phase and a good signal-to-noise ratio. And then to use the regression method to get the Cq values.