You can look up the different forms (rules) of fictional time travel. There are quite a few; but they can be boiled down to three main types:
1. “Can’t change the past.” E.g., time traveller goes back in time to kill Hitler. But at the crucial moment, his gun jams. Hitler lives and history unfolds as we know it. (This is hardly ever used because it rarely makes for interesting drama.

)
2. The so-called “closed causal loop” or “predestination paradox.” Like #1, this results in history remaining unchanged. But the added, mind-bending wrinkle is that that very
act of time travelling into the past
creates the history we know or come to know in more detail. A good example of this is the Heinlein short story
All You Zombies (made into the recent movie
Predestination). The archetypal predestination tale goes all the way back to
Oedipus Rex. By trying to
avoid his prophesized destiny, Oedipus actually
causes it.
3. With “multiple timelines” or “alternative realities,” the time traveller can make all sorts of changes. Classic example is Bradbury’s
A Sound of Thunder. (During a trip back in time to do some dinosaur hunting, the hero accidently steps on a butterfly — which causes dramatic changes to his present-day reality.) Implicitly or explicitly, the original timeline/reality remains unchanged. However, the act of time travel has caused a splitting or branching of reality and history. Whether the time traveller is stuck on this new branch or can (somehow) transport back to the original branch is at the discretion of the author. S/he makes the rules.
In the Arrowverse, it seems most of the time travel shenanigans use #3.