The key is Flashpoint, which affected the timelines of all versions of Barry. In the original timeline, Savitar was a time remnant. In this timeline, he lives in a closed time loop. Let me explain.
1) In the future of the pre-Flashpoint timeline, a villain called Savitar appeared in 2021. Barry created time remnants of himself to fight him, and they imprison him in the Speed Force. One time remnant survived, and was shunned away by the team (including Iris). Driven by envy over the other Barry's life, in the years that follow he becomes progressively mad.
So he travels back in time, acquires the philosopher's stone and has the crazy idea of becoming the very same villain that he helped fight, hoping to change his fate and trap Barry in the Speed Force instead.
However, that attempt was futile to begin with, since the Speed Force exists outside of time, so he was destined to become trapped in it, which is what happens.
2) Flashpoint was a massive distortion to Barry's life since he creates a timeline in which he doesn't even become the Flash! Savitar Barry senses this from the Speed Force. Now he can't possibly have an origin within the Flashpoint timeline, yet his existence is protected by the fact that the Speed Force prison exists outside of time.
3) Eventually, Barry changes his own past again. As of 2016, Savitar is free to craft his own origin within the new timeline. However, oddly enough, he won't have an origin, this time he will exist in a closed loop: Freed from the Speed Force in 2017, he kills Iris to make this Barry suffer as much as he did. Then he will travel back in time to acquire the Philosopher's Stone once, create the Savitar cult, then in 2024 he will fight Barry, who traps him in the Speed Force from which he escapes in 2017.
The memories of him as a time remnant, and his driving motivation, come from the original timeline which has now changed. Slowly, those memories would've been replaced like Barry's did during Flashpoint. Yet we never get to see that, because Barry changes the new timeline I just described at the end of the season.
So to answer your question, no, Savitar never really has a consistent timeline. He is basically a relic of a pre-Flashpoint future, whose existence was preserved by the Speed Force.