Your resoluteness is unfounded. I have pointed out specific issues and asked specific questions which you have not addressed. Your "best candidate" as written ($x,y$ being cosets) makes no sense for the obvious reason that none of the operations in your formula is defined on those cosets. If one tries to save it by choosing representatives, it does
not even satisfy $d(x,x)=0$, nor symmetry, and I have no motivation to check the triangle inequality. You might say you mean something else, but the issue is that no-one, including you, can clearly state what you might mean. —
Torsten Schoeneberg 5 hours ago