I'd imagine 121 makes a lot of difference, since this could easily fall into technique city otherwise.......and it could also be very specific to the aircraft, how much stuff you need to do in the event of an engine failure before the wheels touch down (systems and gross weight wise), and could also be scenario specific. NOT 121 and not specifically answering your question, but using my own experience flying mil fighters, I've never heard of such a distinction being made in my community. Totally scenario specific. If the right engine fails (vs the left) in my current aircraft, you might have a lot more to think about and do before committing to landing. We do have very specific and standardized criteria for things like executing a high speed rejected takeoff (we call it an abort), but once you are airborne safely flying and the jet isn't coming apart or imminently crashing, you generally have some time to run through the procedures......it isn't like discontinuing an approach inside the FAF, or even at DH is really that big of a deal or anything we don't regularly train to single engine. Back to scenario/aircraft specific though, there is a lot to be said about landing if it is safe and prudent to do so and making an airborne emergency a ground one. Fully talking out of my lane WRT 121 ops, but maybe useful discussion fodder for the crowd.