There can be more than prop rotation to be considered, e.g. some aircraft have split hydraulic systems with the Landing Gear on only one. When the engine driving that system fails the gear will be retracted only by the auxillary pump and therefore performance is predicated on gear retraction time in this worst case. Therefore that engine is declared as 'critical'.
In point of fact the only effect is probably an undeclared bonus if flying on the non-critical engine during the retraction segment. The matter of critical engines is probably only academic.