I suspect that the mapping cocone of a map of cochain complexes should be related to the mapping cone by the suspension or translation: $$\mathrm{Cocone}(f) = \mathrm{Cone}(f)[-1].$$
That is, in degree $q$, we should have $\mathrm{Cocone}(f)^q = \mathrm{Cone}(f)^{q-1}$.
I've picked up most of my homological algebra from Kashiwara & Schapira, but they don't discuss the mapping cocone (only the mapping cone), and neither do any of the other sources I've looked through.