Even "Any move except one from the territory being attacked" is too broad. If the territory being attacked attacks the support AND DISLODGES IT, then that support IS cut.
For example, England has Army Ruhr, Army Holland. Germany has Army Kiel, Army Munich, and Army Berlin. Holland attacks Kiel (supported by Ruhr). Kiel attacks Ruhr supported by Munich and Berlin moves to cover Kiel.
In this situation, Kiel's move against Ruhr succeeds, dislodging Army Ruhr. Its support is CUT. This means the Holland move is unsupported, and it will be bounced by Berlin.
BUT. Add French Army Burgundy, which supports the hold of Ruhr. Now Ruhr is not dislodged. In this case, Kiel is attacking Ruhr, and Ruhr is supporting an attack into Kiel, so its support is not cut. Holland's move succeeds, Kiel is dislodged.
So here we see that if you're attacking a territory that's supporting an attack against you, you CAN cut support, but you have to dislodge it in order to do so.