I use this framing: If I make 100 copies of myself so that I can accomplish some task in parallel and I'm forced to terminate all but one, then all the terminated copies, just prior to termination, will think something along the lines of, "What a shame, I will have amnesia regarding everything that I experienced since the branching." And the remaining copy will think, "What a shame, I don't remember any of the things I did as those other copies." But nobody will particularly feel that they are going to "die." I think of it more as how memories propagate forward.

If I forked and then the forks persisted for several weeks and accumulated lots of experiences and varying shifts in perspective, I'd be more prone to calling the forks different "people."