I'm almost 100% certain that no two connections are internally connected on those cartridges. When I was working on my controller, I derived that info from JBC patents, which describe their cartridges, and the heater is in series with the thermocouple (which is something that is correctly stated by some people on various websites, while there's also a lot of crap over there.)
So given the thermocouple is in series with the heater, what would externally appear as a short is actually the thermocouple itself. If you short those connections externally, then you'll just short the thermocouple.
Now some may have tried just using two wires, which would be the end of the thermocouple and the end of the heater, while the mid point would be just left floating. That should kinda work, although I wouldn't recommend this, as the heater in series would add some noise to the temperature measurement, and keep in mind the temp coeff on those is pretty low (about 10µV/°C IIRC, so that's about 3mV for 300°C!)
Looking at the diagram shown by the OP, the outter shell seems to be left unconnected. It's normally grounded. I don't know how the shown controller works, we have no schematic, so I don't know whether one of the connection is permanently grounded internally in the controller or not, and which one. I can just suspect that the temperature measurement does not work, from the symptoms. We know absolutely nothing about this controller, and obviously the OP should contact the seller first. I don't even know how this controller is supposed to detect what kind of tip is connected, since certainly they do not all have the same temp coefficient.