We have four overt non-Observer time travel acts in Fringe on which to base some conclusion. This is not a lot to go on. The difficulty is that we as viewers have access to more information than most of the characters. Most complaints of inconsistency come from viewers unable to reconcile their godlike view of the Fringe universe with the events as the characters in the universe understand them.
If the Oslo doctor basically created the Observers, then the invasion would have happened. The latest possible point to avoid this future is when the invasion would have actually happened; afterwards, the Observers would guarantee it happened, so this is the last possible point to change anything. For this invasion to not happen, someone must stop the doctor. Walter stopped the doctor. So Walter cannot both exist in the past and arrive in the future. So Walter disappeared. It would be an open question whether Olivia and Peter would remember anything about Observers. I would guess "no." Either the past, or their memories of the past, would not contain Observers at all; or, if they did, they would not be the Observers sent back for the illicit purpose of staging an invasion. Some evidence of this is the fact that the Peter-less timeline had Olivia's past be different because e.g. Peter wasn't there in the field as a child, which led to a different outcome for her stepfather and her emotional stability. I would note that I do not believe September (Donald) going through to the doctor would have erased the observers. Possibly that was tried in several loops we never see. Donald could not exist in a future and a past which didn't lead to that future. Walter could—and he could bring the Observer child with him—but only if he disappeared. When Walter disappears, to others the Observer child seems uncaused, but he isn't uncaused, Walter is there, though he reports on a past that doesn't exist (which is why he doesn't exist in the past that does). If Donald attempted to bring the Observer child through, then either both Donald and the child would be uncaused or Donald could not succeed in the plan and the whole loop would continue (it appears the child knows this already).
A similar thing happened when Peter turned on the machine, linking universes. Peter used it to destroy the alternate universe. In the future, Peter found out this would destroy both universes. They built the machine and sent it back in time. But the device they built allowed Peter using it to see the very future they hoped to avoid. In doing so, Peter avoided that fate. Peter could not both be in the future where the machine is created and in a present where the machine never destroyed universes, so Peter was erased at the latest possible moment, namely, when he turned on the machine and made the bridge. That is, we the viewers know this is the cause of the machine. In fact in the timeline itself Peter died in the past. The confusion is that we were at a particular point in time when the timeline was altered. We the viewers exist outside of time in Fringe so our past is not overwritten.
So the paradox is resolved: people cannot exist in the future and in a past that didn't lead to that future. (Which is why the plan to defeat the Observers works.) In both cases, one person in particular was responsible for the timeline being averted and so that person had to cease to exist. These people's disappearances appear necessary so that there are no uncaused events: the existence of September's son, and the existence of the machine. These only appear uncaused for people on the altered timeline. The original creation of the machine itself is likely pinched off from understanding. I would guess that even without the machine the universe was being destroyed, and a future Walter built that machine in order to destroy the alternate universe before it could cause a collapse, but that still caused a collapse and had to be adjusted again, leaving the machine appear totally uncaused. (So long as the universe was going to collapse, Peter and Walter would build the machine, so there was nothing to "fix" and Peter would continue to exist.) But it is only uncaused because we're in 20XX and not deep in the past to talk to "The First People" who could report the cause to us. (And the characters are not in the future where Walter talks to the doctor to speak to him in order to know the cause of the Observer child.)
And this is why Peter disappeared, but not the First People, because the First People were necessary for any future to exist but due to the time loop caused by the machine itself the only future Peter could exist in was one which the machine would destroy the future. If that future didn't happen, Peter couldn't exist, since Peter was the cause of that future. And that future was going to happen because of Olivia and Peter's relationship. And that happened because Peter was in this universe. So Peter in fact died at the lake.
Analogy: it's like you put a number in your calculator, and hit cosine over and over. Regardless of where you start, you end up at the same number. Eventually you can't even tell what number you started with anymore. In the end this fixed point appears uncaused and the original number disappeared. Sometimes in Fringe we're stuck with the consequences of this directly, sometimes we get to see the last iteration before the fixed point converges. Truly, we don't know how many plans Walter and September tried before the fixed point of Walter going to the future was found. On some loops they could have tried other time travel plots, perhaps one of them leading to Bell's discovery of the stasis runes for example (another truly unexplained point, even to September).
At certain key points it seems there is only one way for the future to continue; if it is stuck in a loop, iterations will continue until the future exists. There is always a future, therefore the timeline will always converge to a future. We are lucky in Fringe to see the convergence, rather than only what happens after the convergence.
Another point in favor of this idea is the episode where Peter Weller tries to save his lover, and to us watching appears to actually cause the very death he hoped to avert. Given the way we see time travel operate with regards to Peter and Walter, it's likely that Alistair (Weller) had not in fact caused her to die originally, but Alistair could not both invent time travel and live on with her, so he had to be erased, and the latest possible moment for that was to actually have him cause the accident on the "last" iteration, leaving a seemingly uncaused death (but only because Alistair was not there to explain it to the characters).
Finally we have the episode where Raymond Green is trying to save his wife using her mathematical equations to build a time machine. The act of him doing so was causing a time bubble to destroy the timeline, presumably because the technology in question was not the "right" way to time travel. Because the overall timeline was not altered there was no inconsistency and there was nothing to repair. Instead the fixed point was merely the apparently uncaused creation of the time machine itself (since the notebook had no legible equations in it). But maintaining the uncaused-cause of this notebook actually required no one exist in multiple places (Raymond and Peter were "safe" knowing the origin of the machine), so the overall consequence of this fixed point didn't have anyone disappear.