One of these variants is "when there is no runway left over the nose, we are going to continue."
Well reasoned argument throughout the post. However one has to be very wary of the high-lighted statement. As part of the Essendon Dove accident investigation, ATSB arranged for trials in Canada to take place which measured the amount of runway used for a Dove to lose an engine shortly after lift-off then bunt over and land straight ahead on the remaining runway length. I don't know the figure off the top of my head but I do recall it was surprisingly long. I recall the test pilot allowed a few seconds of decision time uncertainty before closing both throttles to get back on the ground. Fine with an 8000 ft dry runway in front of you but not so good if the surface is wet.
It doesn't make sense to blindly decide to landing ahead on the runway when neither the ME instructor nor his student have practiced such a dangerous course of action. With long runways, by the time the pilot loses sight of the runway ahead which could be at an altitude of between 100 to 500 ft assuming things are normal up to that point, his airspeed should have been well beyond safe climb speed depending when the engine failed. At night with only runway lights for guidance it is difficult to know exactly how much runway ahead is available as it disappears under the nose in the climbing attitude.
To blindly use the "advice" of the disappearing runway view as a decision point, is unwise - as is the other spurious advice of deliberately delaying gear retraction until you gauge you can no longer land safely on the remaining runway length. The drag caused by extended wheels means it takes longer to accelerate towards safe single engine climb speed so the pilot is digging his own grave for no good reason.
The key is to retract the landing gear without delay after lift off and accelerate towards safe single engine climb speed as soon as practicable. Accept there will always be a "dead man's gap" of three to five seconds before the landing gear lever is selected up in which there is no choice but to abandon the take off if the speed is rapidly decaying because of an engine failure.
Providing the pilot is competent at handling the aircraft on single engine and he promptly feathers the propeller on the failed engine, then a one engine inoperative climb away is feasible. Another key to success is prompt lowering of the nose from the normal climb attitude to level flight attitude to prevent further speed decay. It is probable that most accidents involving an engine failure on take off in twin engine aircraft are the result of mishandling by the pilot - not because of aircraft performance limitations.
But to deliberately crash land straight ahead because of a pre-conceived decision point such as losing sight of the remaining runway, smacks of desperation. The subject has always attracted various opinions. For that reason it is important that the instructor gives informed carefully reasoned advice to his student, rather than a rote briefing that attempts to cover all circumstances. At the same time it is the responsibility of the ME student to diligently research the subject of asymmetric handling rather than rely almost totally on the opinions of others -including his instructors. But then, that applies to all facets of flying.
