This would probably be easier to explain with the diagrams I eternally say I'm working on... but anyway.
Firstly, G'raha does not travel back in time "in both timelines", even if you suppose his arrival in Lakeland immediately causes the split. There is only one instance of him time-travelling, and he travels from the dark future to a point before the two timelines diverge.
Secondly - here's where time gets twisty - as much as we say "he came from the bad timeline and changed things so he ended up in the good timeline", that's a simplification from our viewpoint because the story takes place in the good timeline.
Because the diverging timelines would each have a copy of everything that exists at the moment where they split apart, then (at least on the assumption that the split happens sometime between 4.4 and 5.0) that "everything" includes the Exarch as a separate entity to young G'raha Tia.
This is hypothetical and unobservable, but logical. In the bad timeline, the Exarch would still already be long established as the leader of the Crystarium. He would still be desperately trying to reach us - and he would fail. Ultimately all he can do is watch helplessly as events come to pass in accordance with the history he learned in the far future, and wait for the moment where the First is rejoined and everything dissolves into aether - himself included.
(But if he somehow survived and continued to observe events for another two hundred years, he would ultimately see his younger self depart to the First to repeat the cycle.)
It's in this version of events that we find the causal loop. The events of the bad timeline cause G'raha to travel back in time; G'raha's actions do not successfully alter the timeline; the First is rejoined and everything happens in the way that leads back to the future he originally came from.
The split in the timeline breaks us out of that loop (or rather, changing events breaks the loop and forces the timeline to split) and creates a second path that does not need to lead back to the dark future, because that future continues to exist independently and provide the origin point for both the good-timeline and bad-timeline versions of the Exarch.
This example doesn't make sense because it picks a point where there's no reason for things to differ. There's no reason for G'raha to partly follow the same path of actions then arbitrarily stop and do something else. Even if the timelines had split at that point (which I don't believe they had), both versions of G'raha would carry the same intention to work towards saving the WoL. Both versions of events well might play out exactly the same across the two timelines until you reach the variable factor which is either G'raha's success at calling us, or our decision to respond.
This is exactly why I don't like the "time travel itself splits the timeline" idea (even without the events of Alexander seeming to disprove it). For as long as events remain consistent with the version of history that G'raha is working to prevent, it's much simpler to assume there is only one timeline than two running in parallel. They only need to break apart when the current situation is no longer compatible with that previous version of history.
I think it relies specifically on G'raha's actions since he's the only person with pre-knowledge of what will happen (or did happen from a future perspective) and what needs to be changed. Everyone else just acts based on the present situation. G'raha is a variable not because he "wasn't there before" but because he alone knows what's coming.