I think having an explicit box, which allows for two or more simultaneous copies of you to exist and look at each other, is pretty important. Just being created and destroyed in the normal course of things, when everything looks normal, doesn't have the same impact.

My interpretation is that MWI says precisely nothing about you at branch points, because you don't die there - or rather, I don't necessarily consider a single branch point change to be sufficient to make me not 'me'. Further, creating a copy, or multiple copies, doesn't mean anything died in my view.