Okay, so I think the idea I'm working with is that any travel backward in time moves the traveler to a parallel timeline. (Though this isn't necessarily the case with forward travel. Still thinking about that.) So the time/place they leave is actually unaffected by their travel backward in an effort to change the past,regardless of what they themselves believe. They are actually interacting with a different timeline.
First, by way of overview, I'll suggest a viewing order for the franchise-
1. Terminator
2. Terminator 2
3. Terminator 3
4. Terminator Salvation
5. Terminator: The Sarah Connor Chronicles
6. Terminator Genisys
7. Terminator: Dark Fate
The last two on the list would be the only ones I'd say you could swap with each other and still have them work with the framework I'm suggesting. (They essentially happen "simultaneously".)
If I were to try and create a "tree" of some sort, it might look like this(AND SPOILER WARNING FOR ALL TERMINATOR MOVIES & THE TV SHOW):
Skynet in T1 sends back a Terminator to kill Sarah while the resistance sends back Kyle Reese. Both arrive in the past of what is actually a different timeline from their own.
This timeline advances, creating a similar but different Skynet Future. In this timeline(which audiences do not actually see), Skynet sees the threat of John Connor, and is aware of a failed attempt from another timeline's Skynet (meaning Skynet alone is aware of the existence of parallel timelines). So this Skynet sends back the T-1000, while the resistance sends back the reprogrammed T-800, which both arrive in the new and previously unseen "T2 present day" timeline.
In this new timeline, Judgment Day is delayed, but still eventually arrives. And this timeline's Skynet sees the threat of John Connor, unsuccessfully stopped by attempts from two other Future timeline Skynets (in T1 & T2), and so it contributes its own efforts by sending back the female Terminator while the resistance again sends back a reprogrammed Arnie. Yet again, both arrive in a new timeline, that of "present day T3".
Is this starting to make sense?
The "Present Day T3" timeline continues, resulting in Judgment Day at the end of the movie. This same timeline is preserved and is featured in Terminator: Salvation.
Off-camera (in my mind) in Terminator Salvation, Skynet sees that John Connor is a threat, and so sends back the terminator(s) seen in The Sarah Connor Chronicles, which now also brings the focus of the audience to this new "Chronicles Timeline".
At the end of the Chronicles series, John Connor and the "Irish T-1000" are sent to the future, while Sarah Connor stays in the present.
Okay, here's where the real work comes. Granted, I don't NEED to connect all the stories if I'm already committed to a multiple timeline premise. But if I CAN, I'd like to. And with as little mental story writing as possible.
My suggestion is that Off-Camera, following the final episode, John Connor is simply stuck in this future forever. He presumably charts his own path, dying in obscurity or perhaps becoming a leader and hero, while also having the chance to form a truly real romance with the woman that the "Cameron" terminator was apparently modeled after. As endings go, it's a bit dark and bittersweet, but I can still see some closure for John in it. Sarah doesn't get closure back in the present. But that also seems roughly fitting, as she is a character who is haunted and unable to find closure.
So my minimal mental writing efforts focus on the Irish T-1000, in the future, separated from John on her own mission, still determined to thwart Skynet. She evaluates the situation and sees this timeline as a dead end in pursuit of Skynet's destruction. So she makes the radical move of infiltrating Skynet and using future time tech to send back TWO Terminators.
The first is to kill John Connor, which we see play out in Terminator: Dark Fate. "Irish T" rightly calculates that killing John will fully unleash Sarah, who now gives her full attention to combating Skynet, rather than on protecting and training John. This is what actually prevents Skynet from becoming a reality. (Although we still get stuck with Legion in its place.)
The second Terminator sent by Irish T is to kill Sarah as a child(Terminator Genisys). Why would she do both? Linking to Skynet, she would have become aware of the existence of multiple timelines. Multiple opportunities to prevent or destroy Skynet. So she also targets another timeline, sending back a Terminator to kill Sarah as a child, knowingly tipping off the resistance at the same time, so that they could send back their own reprogrammed "Pops" terminator. Again, this is an effort to shake-up Sarah's timeline and keep her from being distracted with raising/protecting a child, so that she can fully commit to fighting Skynet. This plays out in the Terminator Genisys story. (We don't need to be concerned about what timeline the opening scenes of Genisys emerge from. This is just a "similar but different" timeline with no direct timeline continuity with previous stories.)
Now I have very little familiarity with the Sarah Connor Chronicles. I watched and loved it when it aired, but have only watched the last half of the final Ep since then. So I am likely missing some important details and motives related to "Irish T" and her "son" John Henry.
With that in mind I invite your help to correct or further flesh out what I'm working on here.
Otherwise, I hope you enjoy one hopeless nerd's efforts to do something that is completely inconsequential to real life while he begins to enjoy some vacation time!
Merry Christmas!