I would agree with Richard that the writers probably didn't think this through in great detail and just were concerned with a story that made enough sense for people to follow the narrative. I also agree that any explanations you can come up with to account for all the events of the movies would have to be fairly hand-wavey ones. Still, I wouldn't go so far as to call this issue a "contradiction". If you're nerdy enough to want to think about BTTF "temporal mechanics" in detail, I think it is possible to come up with some basic rules for how time travel works in the BTTF universe that, while a bit arbitrary and kludge-like, are at least consistent with everything we see on screen, and with comments Gale and Zemeckis (mostly Gale) have made offscreen about how they imagine time travel to work in their movies.
Rule #1: There is a single timeline which can get re-written, with the "changes" happening in a sort of second time dimension, or "meta-time". Thus we can talk about "earlier" or "later" versions of the timeline with it being understood we mean "earlier in meta-time" or "later in meta-time". If it isn't clear what I'm talking about here, or you don't see why BTTF should work this way, please read this answer I wrote to another BTTF question. The chart Axelrod posted in answer to the same question, showing a "sequence" of different versions of the timeline created by various time jumps (which according to this rule should be interpreted as a sequence in meta-time), is also helpful:
Rule #2: If a time traveler makes a trip in time to some date and place X, and this trip didn't occur in earlier versions of the timeline, then their memories and physical appearance when they materialize at date/place X are "burned in" to all later versions of the timeline. For example, when we see Marty first appearing in 1955 at the Twin Pines Ranch, that's the first version of the timeline where that occurs, and the Marty who appears there is the one who has memories of growing up with a wimpy father who was still being pushed around by Biff. Thus, according to this rule, in all later versions of the timeline, regardless of what the Marty of the new timeline experiences growing up before 1985, the Marty who appears at the Twin Pines Ranch in 1955 will still remember growing up with the wimpy father.
Rule #3: Apart from the burn-in effect above, the timeline tries to preserve events as best possible when history is changed, in particular preserving the details of the events leading up to a trip through time which happened in previous versions of the timeline. So in timeline #2 from the chart above, Marty should grow up with an altered family with happier and more confident parents, but the self-preservation effect would ensure he still ends up meeting Doc, and still ends up saying and doing the same things in the mall parking lot (now the Lone Pine Mall, not the Twin Pines Mall as in the previous timeline) on Oct. 26 1985, and still ends up making a time jump from 1:35 AM on Oct. 26 of 1985 to Nov. 5 of 1955. But according to Rule #2, since this is a trip that already happened in prior versions of the timeline, upon arrival in 1955 his memories instantaneously reset to those of the Marty who grew up with a wimpy father. So, there is no need for this trip to spawn yet another new timeline where a different version of Marty appears in 1955. I think of Rule #2 as conceptually different from the self-preservation effect, since that effect is more like a gradual guiding of events back on track, whereas Rule #2 is an instantaneous reversion of memories and possibly physical appearance (in case the Mary of timeline 2 had a scar in a different place or something).
Incidentally, as Richard said in his answer, Gale and Zemeckis do discuss the idea of a self-preservation effect in the Official BTTF FAQ--Richard quoted from question 1.12 which mentioned the effect in reference to the issue of the two Jennifers meeting, but they go into a little more detail about what they mean in question 1.19 where they say:
There's a theory (we like to call it the "Self-Preservation Instinct of the Space-Time Continuum Theory") that says that the continuum is always trying to keep itself "on course," and when things happen to change it, it always tries to correct itself. It is much like a river, which tries to keep its overall course. Although earthquakes, fallen trees, floods, or other circumstances might disrupt it at points, the river would cut a new channel so that it would end up back at the same place.
One question that might crop up about this: what happens if the timeline is too badly altered to ensure that a certain time trip can happen "naturally"? For example, in the "1985A" version of history where Biff is rich and powerful (timeline 4 and 5 on the above chart), Doc has been put in a mental asylum, so unless he escapes and gets the money for equipment, he presumably can't build a time machine. Also, Biff says at one point to Marty "You're supposed to be in Switzerland you little son of a bitch!", indicating that the Marty who grew up in that timeline--let's call him Marty-A--was supposed to be abroad on that date. So it seems unlikely that the self-preservation effect could have ensured that Marty-A would meet up with Doc-A at the location of the Lone Pine Mall (if it even still existed in this timeline) and travel back to 1955 in a DeLorean time machine provided by Doc-A. In general we know the self-preservation effect can't be perfect since if it was there would be no danger of a time traveler erasing themselves from existence by changing the past.
One possibility here is that the "laws" of the BTTF universe would allow two alternate versions of Marty at the same biological age to coexist in a single timeline (with Marty-A still in Switzerland while the Marty from timeline 1 confronts Biff). Another is that Marty-A would simply be forcibly pulled out of time on Oct. 26 1985 of this altered timeline (perhaps along with a DeLorean sitting in a garage somewhere) and would spontaneously jump to 1955 with all memories and physical appearance reset to those of the Marty of timeline 1 upon arrival. This would be a sort of brute-force method the universe could ensure there is only a single version of Marty at each biological age. (Biological age seems to be important in the BTTF universe, since the answer to question 1.8 in the official FAQ mentions they had wanted to have old Biff disappearing upon returning to 2015, because in the new altered history he had died at a younger age than the old Biff we saw. Note that by 'biological age' I mean the precise age rather than just how many years old someone is, so for example when Marty watches his younger self in the parking lot at the end of part 1 or playing on the stage in part 2, the two versions of Marty in these scenes wouldn't be the same age since one is a few days older than the other. The idea here is that the 'laws of time' want to ensure that for each day of a person's life, each hour etc., there is only a single version of them at that age anywhere in the timeline.)
If the brute-force method seems weird, note that there actually is precedent for it in this early BTTF script draft written by Gale and Zemeckis. In this version of the script, the changes Marty made in the past (1952 in this version) had more drastic effects on his present--when he returned to his own time he found a world filled with Jetsons-style retro-futuristic technologies, some labeled as being products of "E. Brown Enterprises", since Doc's experiences with Marty in 1952 led him to abandon time travel as too dangerous and focus his inventive skills on other things. So if the Doc of this timeline never built a time machine, what happened to the Marty who grew up in this timeline? Doc explains that he just disappeared on the appropriate date:
MARTY
But if you didn't rebuild the Time Machine, how did I go back in time in the first place?
PROF. BROWN
According to your girl friend, Suzy Parker, you and she were at the movies. You went to the restroom, and you never came out. Obviously, you stepped through an inter-dimensional time warp, created by the original operation of the time machine.
Finally, we need one last rule to explain situations where the whole world apparently changes around a time traveler by the ripple effect at a moment when they themselves aren't making use of a time machine (again see my other answer about meta-time for a discussion of this). This brings us to:
Rule #4: Suppose on some date D a time traveler #1 makes a trip back in time, but there is some other time traveler #2 present on the same date D. And suppose we're considering the very first timeline where A makes this trip on date D. In that case time traveler #2's memories on or after date D also get "burned in" to the timeline. This means that the versions of time traveler #2 in all later versions of the timeline have their memories "overwritten" by the memories of time traveler #2 in the original version of the timeline where time traveler #1 first made the trip.
For example, in part II old Biff (time traveler #1) makes a trip back from 2015 (date D) to 1955, while Marty (time traveler #2) is in 2015 as well. Since Biff's trip to 1955 creates a new timeline where he got rich gambling, 2015 should transform to 2015A, and the answer to question 1.9 of the FAQ indicates this happens around Marty (and Doc and Jennifer and Einstein) at around the moment that old Biff returns to 2015. So according to rule #4, another way of looking at this is that Marty's memories at around the moment old Biff returns get "burned in", so they must re-emerge at that moment in all subsequent timelines. So even if Marty's experiences in the 2015 of the new altered timeline (timeline 4 on the chart) had been a little different up until that point (for example, he might have seen the 2015 version of Biff's Pleasure Palace earlier in the day), at that moment they would be replaced with memories of his original trip to 2015 (illustrated by the lighting bolt from timeline 3 to timeline 4 on the chart).
Also, if the neighborhood he was in at that moment had any visible differences between the two timelines, then suddenly having the memories of being in timeline 3 up until that moment but now finding himself standing in timeline 4 would make it appear that the world had shifted around him. So the idea that the world "transformed around" Marty, as explained in the FAQ, would still be accurate as the subjective perspective of the time traveler. Whereas my description above is more focused on what would happen from the perspective of a "native" of timeline 4 (or what be seen if you had cameras set up around town in the 2015 of timeline 4, and then after old Biff returned you checked what these cameras had recorded of Marty's behavior throughout the day).
And what if old Biff hadn't actually returned to 2015 from 1955, or perhaps had returned to some date prior to the date D he left? In this case I think it would be Marty's memories right at the date D that would get "burned in" and would overwrite the memories of other versions of Marty visiting 2015 in later versions of the timeline. This is suggested by what happens later in part II, when Marty sees Doc in the hovering DeLorean get struck by lightning, and then almost immediately is greeted by a Western Union guy with a letter from Doc in 1885. According to the chart, Marty has just experienced a transition from timeline 6 (where there was no Doc in 1885, and thus no Western Union guy driving towards Marty with a letter for him) to timeline 7 (where Doc had been present in 1885, and the Western Union guy would remember the letter having been around for a long time, and probably had been driving a while before finally reaching his destination). Perhaps if some non-time-traveler from timeline 7 had been spying on Marty at that time, he might have observed that Marty noticed the headlights from the Western Union truck approaching in the distance while Doc was still hovering in the sky, but then at the moment the lightning struck and Doc disappeared, Marty's memories would revert to those of timeline 6. So if the spy had bothered to ask, Marty would claim not to remember seeing any approaching truck before the lightning strike.
I think these rules are sufficient to be consistent with everything we see on screen and avoid major plot holes like why the Marty in the Lone Pine Mall parking lot doesn't change history yet again when he jumps to 1955, despite him having different memories than "our" Marty. And you'd need some additional rules to deal with inanimate objects that have been brought through time, like photos and newspapers and matchbooks, since they don't seem to work quite like human memories (a point that's brought up in the answer to question 1.19 of the FAQ), but the OP's question wasn't really about that, and this answer is long enough already!
