This is not a true answer so much as it is a guideline for what an answer should be. It's my best attempt at an answer, but there's a gaping hole in the Salvation change. I'm also not sure if the list of discrepancies I address is complete or not. I've relisted it as an answer instead of being part of the question at the community's request.
The Terminator
- A T-800 is sent back in time to assassinate Sarah Conner. Kyle Reese is sent back to save her. This is the original timeline. While there are numerous effects of the two time travelers' presence (Sarah's pregnancy, numerous deaths), we learn by the end that these were not changes to the timeline, but rather that the orignal timeline already included this visit from the future, and the machines created the very future they went back to prevent. No changes to the timeline.
Terminator 2: Judgement Day
- Sarah, John and the T-800 destroy the remains of the original terminator and leave no remains of the two that came back to 1993. Because the existence of Skynet and Terminators was predicated on having future technology to go by, this seemingly prevents Judgement Day from happening. However, they were only successful in delaying the inevitable. The original origin of Skynet is prevented, delaying Judgemnent day 10 years.
For this explanation, I argue that the two terminators coming back to 1993 was not a change in the timeline in and of itself. While Kyle states "It's just him and me" in the first movie, we have every reason to believe he came back as soon as possible after learning the first T-800 had gone back (as is the case in subsequent timelines). In her opening narration in Judgement Day, Sarah states that the machines sent 2 machines back to attack them. It is likely that the machines sent the T-1000 back shortly after the T-800 (possibly from a different location), or at least that the resistance didn't learn of the second attempt until after Kyle had departed. Either way, the films support both attacks being part of the original timeline. However, in the original timeline, though John lived, Judgement Day wasn't prevented. (Perhaps John didn't realize he could give the T-800 orders, and it took John out of Los Angeles instead of rescuing Sarah.)
Terminator 3: Rise of the Machines
- The TX succeeds in eliminating some key resistance members before being destroyed by the T-800. This is the only confirmed change to the timeline created in Judgement Day that occurs during the events of Rise of the Machines. In the future this film's terminators come from, Judgement Day and John meeting Kate happened just as they do in the film. While we don't see any direct consequences of this, there are changes observed in the next film which must either be a consequence of this change or the one listed above for Judgement Day. Namely, The resistance (or at least Kyle Reece) now prefers to engage terminators in daylight rather than at night.
Terminator Salvation
- Skynet changes something offscreen. Yes, this is a terrible explanation, and an answer hoping to be accepted can't contain this hand wave, but that's part of why I'm asking the question. There is no time travel depicted in Salvation. Yet, the details of John and Kyle meeting change from Salvation to Genisys. More importantly, something prompts Skynet to make the most radical change to the timeline yet in Genisys. (If nothing changed between Rise of the Machines and Genisys, there would be no motivation for the events of Genisys to occur. John meets Kyle at a different time, and Skynet develops a plan to re-establish itself after its defeat.
Terminator: Genisys
- Skynet and the resistance send Terminators to Sarah as a child. Genisys details this pretty clearly: Sarah's childhood is drastically changed and she becomes an effective soldier much sooner. Much of the collateral damage done in 1989 is prevented. Pops's continued presence is also responsible for the prevention of Judgement Day in 2003.
- Skynet infects John Conner while Reece jumps through time. This starts a chain reaction. Witnessing a timeline change while time travelling gives Reece an erratic ability to access his own memories across timelines. This in turn enables him, Sarah and Pops to travel to address the new Judgement Day threat in the correct time. Judgement Day is again prevented in 2015, delayed until an as yet unknown time.
