Even with those rights changing hands as often as they have, I understand that any franchise so centred around time travel is going to be inherently problematic in this sense, but my attempts to work out various continuity issues go beyond (and ultimately have nothing to do with) the time travel itself.
As I alluded to when I talked about potential fictional timelines I've considered putting together, my thoughts on all this started with my wanting to create a Terminator Timeline with just two films to worry about, only to discover fundamental incompatibilities when the third film was released. I hadn't considered it much for a while, though--beyond relegating that film to a parallel universe in my mind--until reading in-depth about how such matters are handled by fans of The Transformers. (Robots in disguise need to stick together, I suppose.) Their approach, with its complex but compelling "continuity family" structure, made me revisit this issue and how it's reflected in media franchises like this one and others with frequent tie-ins and crossovers.
With that in mind, it made the most sense to divide the numerous iterations of The Terminator into continuity families based on works that are properly related to one another, so those are outlined below in the order of their original release. Come with me if you want to see:
This is the world of The Terminator, Terminator 2: Judgement Day, and (as far as I'm concerned) T2 3-D: Battle Across Time. Judgement Day is August 29, 1997. John Connor looks like Edward Furlong. This is the core continuity, for obvious reasons, and what most people think of as the Terminator Universe. As such, it is also the continuity with the greatest number of offshoots.
I assume the main timeline is essentially one big causality loop (even after the second film modifies it), though there is a timeline where Judgement Day is prevented (T-Cameron-A, the world of the alternate ending to T2), and there may hypothetically be a timeline (T-Null) where Kyle Reese is not John Connor's father. (I also assume that Kate Brewster does not exist in this continuity.)
Branches include anything based solely on the James Cameron material, where either the first or the first and second movies are known to have occurred exactly as shown onscreen. Some of these include: T-Fortier, the world of the titles from NOW Comics up to Terminator: The Burning Earth (which also happens to be the first professional comics work by Alex Ross); T-Arcudi, the world of the titles from Dark Horse Comics up to The Terminator: Endgame (in which Jane Connor is born and leads the future resistance to victory); T-Miller, the world of RoboCop Versus The Terminator (in which RoboCop is the missing technological link allowing Skynet to be created); T-Stirling, the world of the novel T2: Infiltrator and its sequels by S. M. Stirling; and T-Blackford, the world of "The New John Connor Chronicles" by Russell Blackford, starting with the novel Dark Futures.
This is the world of Terminator 3: Rise of the Machines. Judgement Day is July 25, 2004. John Connor looks like Nick Stahl. Some version of the events of T1 and T2 occurs in this continuity, but the dates in the movie itself suggest that those took place in 1979 (instead of 1984) and 1993 (instead of 1995), respectively. (According to the film, Kate Brewster's existence and importance to future events explains the timing of its version of T2.) In this continuity alone, the T-850's appearance is based on that of William Candy, a sergeant in the US military.
Branches include any of the mutually-inconsistent tie-ins to this movie, such as Terminator Dreams by Aaron Allston and Terminator: Infinity by Simon Furman.
This is the world of Terminator: The Sarah Connor Chronicles. Judgement Day is April 21, 2011. John Connor looks like Thomas Dekker. Some version of the events of T1 and T2 also occurs in this continuity, but its version of T2 takes place in 1997. Kate Brewster appears to once again not exist, but Kyle Reese has an older brother named Derek who travels back in time to protect Sarah and John.
The series itself dealt explicitly with several branching alternate timelines, with plots about time travellers going back to before the events of T1 and even a cliffhanger at the end of Season 2 in a future where no one has ever heard of John Connor. However, the series had no official tie-ins.
This is the world of Terminator: Salvation. Judgement Day appears to be in 2004. John Connor looks like Christian Bale. This continuity shares certain traits with T-Mostow (such as the existence of Kate Brewster), but seems to occur in a world where only a version of the events of T1 happened, since John Connor has apparently never met a T-800 before the film--he listens to tapes of his mother describing the future (as described to her) rather than personally reflecting on his direct experiences in T2 and/or T3.
Branches include The Machinima Series and the mutually-inconsistent tie-ins to this movie, such as Cold War by Greg Cox, Trial by Fire by Timothy Zahn, and The Final Battle by J. Michael Straczynski.
This is the world of Terminator: Genisys. Judgement Day is in 2017. John Connor looks like Jason Clarke. A T-1000 is sent back in time to 1973 in an attempt to kill Sarah Connor, but she is saved by a T-800 sent back to protect her. (This appears to happen in lieu of the events of T2.) "Pops" subsequently raises Sarah to prepare for Judgement Day, with the two of them stopping the original T-800 in 1984 and intercepting (a version of) Kyle Reese during what would otherwise be the events of T1.
There have not been any sequels or tie-ins to Genisys at this point, but behind-the-scenes materials suggest that the T-5000 played by Matt Smith is a manifestation of Skynet from an alternate timeline, so the film's intended sequel(s) may yet connect this continuity more explicitly to T-Cameron or even unify these continuities into a greater multiverse.
All of this doesn't quite rise to the convoluted level of the Universal Streams explanation within the Transformers franchise, but it's getting there (even within the fiction itself), and I'll admit it shares certain similarities. For instance, many titles agree completely with one or more of these timelines, but create dead ends of continuity not worth outlining in detail. (Some of the crossovers I didn't mention, such as Aliens Versus Predator Versus The Terminator or Superman Versus The Terminator: Death to the Future, are good examples of this--especially since they are dead ends in their own continuities as well.) If you're ever looking for further and further deviations in the seemingly straightforward story of machines from the future and the humans trying to stop them, however, I'm sure they'll be back.