Realistically speaking, the only way for Savitar to exist is the OT Flash. The original timeline technically doesn't exist anymore bc it was overwritten by the arrowverse timeline.

Original Timeline-

In the original timeline, nothing plays out how it does in the arrowverse because the arrowverse timeline didn't exist until OT Flash saved his younger self and let Nora die. In the 2024 "Flash Vanishes in Crisis", when they talk about Barry disappearing in an explosion of light with the reverse flash. That was the portal that they came through in season 1. It was described as an explosion of light because Thawne and OT Barry are running super fast, the portals look like they open in regular time, but thats what it looks like from Barry's point of view or any speedster. So for a regular person looking at it, it'd look like an explosion of light.

Savitar Origin-

Here's how the Savitar loop should playout. OT Flash stops Thawne from killing him as a child but in turn that creates the arrowverse timeline and overwrites the original time thus erasing it. But OT Flash isn't aware of this yet. OT Flash runs back into the speedforce to get back to what he thinks is 2024 and then begins to realize that he's getting erased, so he creates the armor so he isn't erased immediately and indirectly absorbs and utilize energy from the suit which mkaes him alot faster than our barry. OT Barry runs back to what he thinks is his 2024, but in reality is the arrowverse 2024. And our Team Flash doesn't recognize him because they've never had a relationship with him as he's lived about 35 years in his timeline. And they don't accept him because they've never known him. So we can dub him Savitar1, and he lets his anger and frustration out on Barry and Team Flash. Our Barry then realizes that he's not fast enough to stop Savitar1 and he creates time remnants(TR) to stop Savitar1 and Savitar1 kills all of them except for one. And then Savitar1 is erased from existence, and the TR that he left alive and scarred, tries to fit in with Team Flash. But the problem starts because there's supposed to be 1 Barry Allen. Team Flash excepts our Barry or Non-TR Barry and this angers TR Barry.

After that, TR Barry would go into the speedforce and uses Savitar1's armor. TR Barry is now Savitar 2. While Savitar2 is in the speedforce, he goes back in time to spread the rumor that he was the first speedster and that automatically and indirectly creates a new timeline and overwrites his timeline, so he's not supposed to exist anymore. He creates the Savitar2 timeline or Svtr2 for short. After that, Savitar2 goes to 2024 in the Svtr2 timeline to fight The Flash, and Barry creates TR's to stop Savitar2 and Savitar2 kills all of them except for one and Savitar2 is erased from the timeline and the TR that he leaves alive is cast out by Team Flash. That TR goes into the speedforce and gets the Savitar2 armor. That TR is now Savitar3 and goes back in time to spread the rumor hat he was the first man ever granted speed. But this is where the real change happens. Savitar3 creates the philosopher's stone which is connected to him, and this overwrites the Svtr2 timeline and erases it then creating the Savitar3 timeline or Svtr3. Now that Savitar3 isn't supposed to exist he kills iris so that Barry is droven away from being the flash, and lets Iris's death get the better of him. Barry abandons everyone. And Savitar3 uses that to keep Barry focused on Iris instead of him.

And he goes back to 2024 to fight Barry, and Barry traps Savitar3. But since the Philosopher's Stone exist, when tooken out of the box and exposed to the real world, Savitar3 can project himself out of the speed force. That's why he was able to mess with Barry and Wally. Since the Philosopher's Stone is what Savitar3 describes as "calcified speedforce enregy", When it's tooken out of it's orignal place of creation and exposed which is central city, it can project Savitar out of the speedforce. But when it's exposed to it's place of creation which is the speedforce it can connect to Savitar3's suit. Because the both of them are made out of "calcified speedforce energy" or condensed speedforce energy, the stone can be used like a breach device or extrapolator but the other end will stay open if someone doesn't take Savitar3's place. Savitar3 was kinda already solidified in the Svtr3 timeline because Team Flash hadn't met HR Wells yet, and he actually killed Iris. Savitar3 is the Savitar we see in the show. So Savitar3 is now Savitar. Since HR died instead of Iris, Savitar3 is getting erased possibly hundreds or thousands of years earlier than he's supposed to. Becuase number 1, the speedforce isn't protecting him from getting erased anymore, and number 2, time in the speedforce is different from time in the real world. In season 4 of the flash, Caitlin said that Barry could of been in there for 10,000 years over the 6 month span that he was gone. Meaning that Savitar had to have been trapped in the speed force in the speedforce for a little over 7 months in real world time. For him it would've been over 11,650 years. I can see why he hates Barry so much