(Yes, I know this topic was started over an year ago)
I didn't know about <PERSON>'s deleted scene where he fades out of existence. Because of that, this has become my point of view:
* In BTTF1, <PERSON> traveled back to 1955-A and changed his future, thus creating a 1985-B where he didn't exist. The future took some time to change, but since he was 30 years in the past, changes took longer to catch up to him so it took almost half the movie for him to start disappearing, giving him the chance to fix things. When his parents finally got back together, he created a new 1985-C where he did exist and also time traveled, so he stopped disappearing.
* In BTTF2, <PERSON>, <PERSON> and <PERSON> traveled to the future. How can their future selfs exist there if they disappeared in 1985? Because the timeline takes time to adjust like in BTTF1. If they had stayed long enough, they would have transitioned/faded into a new 2015 where their future selfs don't exist. 663
Also, any changes they made to their future will be futile because the moment you know something about your future, you're already changing it.
* <PERSON> traveled back to 1955-C and drastically changed the future, thus creating 1985-D and 2015-D (we never saw 2015-D). Again, changes took some time to happen, so when he returned to 2015, it was still 2015-C. He eventually faded into 2015-D in the deleted scene (dead or alive, doesn't matter) taking much less time than <PERSON> in the 1st movie.
* <PERSON>, <PERSON> and <PERSON> were still in 2015-C, but they weren't the ones responsible for the new 2015-D neither they belonged in 2015, so 2015-C only ceased to exist for them when they left in the Delorean. By traveling back to 1985, they entered the new 1985-D timeline created by <PERSON>.
* Since they traveled back 30 years into the past, and then another 30 years into 1955, they had enough time to fix things before starting to disappear. The Delorean didn't exist in 1985-D because <PERSON> was in an Asylum and never created it, but since it's the only tie-in between timelines, it cannot simply cease to exist, but it could still take damage, and eventually was destroyed in BTTF3.
* In BTTF3, <PERSON> and <PERSON> didn't disappear because their future was fixed, they only needed to return to it. <PERSON> was left on the porch in 1985-D and when <PERSON> and <PERSON> shot their way into a new timeline (changing Clayton Ravine to Eastwood Ravine), she transitioned/faded from 1985-D into 1985-E while she was sleeping.
Basically, all the original plot needed to be more consistent was the deleted scene and a better explanation from <PERSON> about the "fading", but that could become too technical for the audience.
(Edited to include all 3 movies) (edited to explain <PERSON>'s existence in the future of BTTF2) 837