The "stick" is the profile you run down the inside edge of the stiles and rails. The "cope" is the matching shape cut on the ends of the rails so the corners close up tight and line up. This is the everyday machined frame for most cabinet doors.
Here the rails have square ends with a tenon (a tongue) that fits into a mortise (a slot) cut in the stile. The joint is held by that tongue-in-slot fit plus glue, so there is no profile to match — you use this when the frame edges are left square instead of shaped.
DoorCut uses the same finished rail and stile sizes for both methods: the rail length is the inside opening plus twice the tenon length you typed in (the way traditional cut lists do it). Picking cope & stick or mortise & tenon just records how you plan to build it — it does not change those sizes. The shop details (which cutters, how deep the mortise, haunches, and so on) are still up to you at the machine.
One honest heads-up: cope & stick has a long track record behind its numbers, but mortise & tenon does not have a legacy reference in this app. M&T is figured with the same blank-size formula as cope & stick — rail length = inside opening + twice the tenon — just with a longer tenon. That gives you the right rectangular blanks if the tenon length you type matches how you actually lay out the joint.
Because M&T is the typical full-size door (and a different animal from a cabinet door), check the cut sizes against your CAD model or a test door before committing stock — especially the rail length and your tenon convention. The math is straightforward, but there is no 30-year app double-checking it for you on this path.