Mido
Version 1
One alternate ending heavily implies that it'll be for purposes of completion, perhaps finishing all the Shrines and dungeons restores the Master Sword to it's full power, thus allowing Link to finish him for good?
While if you don't do that, perhaps they'll have to settle for sealing it.
This notion sort of reminds me of how Star Fox 64's endings were executed. One ending tied up all the loose ends while the other left unfinished business lingering in the air. Could be the same here if then endings are presented in the manner that you postulate. I assume for the sake of streamlining, one of the endings will be the "true ending" of sorts.