Which, if any, version of Terminator 2 is officially canon?

Question

There are three official versions of Terminator 2: Judgment Day that I know of: the theatrical cut, the special edition, and the extended special edition.

The special edition and the extended special edition both include a scene where Sarah and John remove the T-800's CPU chip from its head, in order to reset the switch from read-only memory to read-write memory. This enables the T-800 to gradually get better at both emulating and understanding human behaviour, culminating in its ability to understand why John was crying near the end of the film.

The chip-removal scene is preceded by the following dialogue:

JOHN: Can you learn stuff that you haven't been programmed with? So you can be... you know, more human? And not such a dork all the time?

T-800: My CPU is a neural-net processor; a learning computer. But Skynet presets the switch to read-only when we are sent out alone.

SARAH: Doesn't want you to do too much thinking, huh?

T-800: No.

JOHN: Can we reset the switch?

In the theatrical cut though, the chip removal scene is omitted, and the preceding dialogue is modified somewhat, as follows:

JOHN: Can you learn stuff that you haven't been programmed with? So you can be... you know, more human? And not such a dork all the time?

T-800: My CPU is a neural-net processor; a learning computer. The more contact I have with humans, the more I learn.

JOHN: Cool.

The T-800 subsequently experiences all the same character-growth in the theatrical cut as in the other cuts, culminating in its ability to understand why John was crying near the end of the film. But it apparently does this without Sarah or the young version of John needing to tamper with its CPU.

Given this apparent contradiction between the theatrical cut and the other versions of the film, the question is begged: which, if any, version of T2 is officially canon?

Answer 1

Although I personally believe it is practically impossible to agree on what's canon in a franchise that rewrites not just the past and future, but the actual rules of time travel as it goes, the closest we can ever get to 'canon' for T2 is Word of God.

Although he doesn't use the word 'canon' specifically, it's clear from the commentary on the Special Edition that James Cameron himself considers the Theatrical Edition to be the definitive release of Terminator 2.

I still stand by the release version of the film.

I think we made the right decisions editorially and I think that the dream sequence and some of the other things that we put back in for the extended version are good to sort of show the creative process, but, you know, essentially, making a film is an analytical and a reductive process. You throw away the things you don't need.

Specifically, this next passage (emphasis mine) would strongly suggest to me that Cameron views the Extended versions as snapshots of the creative process in motion, like early drafts of a script.

The special edition is really like selling the donut holes. It's part that was left over when you did the perfect release version of the movie. There are a lot of little scenes that are good. There's nothing wrong with them. It was the same cast, the same skills, the same writer, the same director, all of these things, but it turned out to be slightly less necessary than the pieces on either side of it in the final cut of the movie. So there's certainly a group of people that would be interested in seeing what those scenes were, and what some of the extended canvas might have been, coming from the same actors, the same character.

It's worth noting that, unlike some other extended releases of movies, these Special Editions are not, and have never been, Director's Cuts. In the world of James Cameron, the Theatrical Release is the Director's Cut.

As an aside, the scene you specifically mention with the CPU is the only one that Cameron regrets removing for the Theatrical Release, although I suspect this is due more to the dialogue between John and Sarah being an important part of their respective character arcs.

From the commentary just before that additional scene:

I don't really miss anything but the next scene.

It's worth noting that the removal of this scene doesn't necessarily preclude the chip having to be reset to read/write, it just means that, if it did need to be done, it must have been done as part of the original reprogramming of the T-800 by the resistance before it was sent back, which makes a lot more sense anyway.

Answer 2

Terminator: Dark Fate appears to treat the theatrical cut of Terminator 2 as canon

Terminator: Dark Fate opens with a scene where a T-800, sent by Skynet, successfully terminates a young John Connor, a while after the events of T2.

By the time we see that T-800 again though, much later in the film -- roughly 20 years later, in-universe -- it now goes by the name of 'Carl', has an adopted human family, and runs its own drapery business.

SARAH: Nice family. She a Terminator, too? That's your little Terminator kid?

CARL: His name is Mateo. I met his mother, Alicia, a few months after I killed John.

SARAH: Oh, you don't get to say his name. Ever.

CARL: Her husband had beaten her. He was trying to kill her child. She had nowhere to go. Caring for this family gave me purpose. 'Cause without purpose, we are nothing.

_{Terminator: Dark Fate (2019)}

It also seems to feel remorse over having terminated John, and chooses to help Sarah, Grace, and Dani defeat the Rev-9, ultimately sacrificing its own life, "for John" and the human race.

SARAH: Touching story. Does it have a point?

CARL: While raising Mateo, my son, I began to understand what I had taken from you.

GRACE: Wait. You grew a conscience?

CARL: The equivalent of one, yes.

SARAH: It's an infiltrator. It's lying.

CARL: When my mission was completed, there were no further orders. So for 20 years, I kept learning how to become more human.

GRACE: So what about the texts?

CARL: When chronal displacement occurs, there's a shockwave through time, measurable before the event.

SARAH: That's how, not why.

CARL: To give you purpose, Sarah. I thought it would bring meaning to your son's death.

[...]

CARL: Do you believe in fate, Sarah? Or do you believe that we all can change the future, every second, by every choice that we make? You chose to change the future. You chose to destroy Skynet. You set me free. And now, I'm going to help you protect the girl, because I choose to.

_{Terminator: Dark Fate (2019)}

This is clearly a T-800 which had grown far beyond its original programming, and it did so without any humans tampering with its CPU. That only makes sense if Terminator: Dark Fate is a direct follow-up to the theatrical cut of T2, specifically, rather than the other versions of the film, where the switch on the CPU had to be reset in order to enable that type of character-growth.

Terminator: Dark Fate is also the only film in the franchise, other than the first two, that James Cameron had any direct involvement with. And while he's but one of six different writers credited with working on the script for Terminator: Dark Fate, the following quotes from interviews with him suggest it was largely his idea to explore how a T-800 would grow and evolve, if left to live among humans for 25 years.

JAMES CAMERON: I had some ideas for Terminator sequels that had lain dormant for a long time. But I knew that, of course, Arnold -- as an actor, as a human being -- was getting older, and so the later you come back to that -- with him in the role -- you have to deal with it. But I thought, "Let's make that an asset." Alright, so we're going to cut to a Terminator that was sent back in time, to some time in the '80s or '90s, hit his target, and had no further purpose. And was waiting for orders from a future that had ceased to exist. Because when Sarah changed Judgment Day, and her actions led to the demise of Skynet -- or the non-emergence of Skynet, really; it never existed in the first place -- the Terminator had no boss. He's got a learning computer; his central processor is a neural-net chip, it's a learning computer. He's given a mission, not only to execute a specific mission, but to blend into human society so he can operate, so he doesn't wind up with all the forces of human society against him, hunting him down and destroying him. So he can just kind of blend in and be an infiltrator, that's the concept, right? What happens if you leave that learning computer switched on, learning, with the goal, the programmed goal of becoming human, for 25 years? We saw him in movie two, over the space of two or three days, already starting to change, already starting to become, you know, warmer, funnier, use language better, right? What happens in 25 years? So, that was, I think that was the challenge that I brought to the writer's room. I said, "Let's do this. This is gonna be radical. This is gonna be cool. We're gonna create a Terminator who wants to be human."

_{TERMINATOR DARK FATE Full Cast Interviews}

JAMES CAMERON: It was something that I thought, you know, "Let me explore this. Let me see if there's something there to be said that hasn't been said." Um, you know, I wasn't caught up in the nostalgia of it. I didn't necessarily want to go back and recreate past glories, you know; what new can be said? And I think we came up with some fresh ideas. The one idea that did propel me was this idea of the Terminator evolving to a new level that we haven't seen before. And by the Terminator, I mean, not the new guy -- who's obviously a new creation, new powers, that sort of thing -- but the Terminator we know and love, which is Arnold's Terminator, the T-800. Now he's a character that just keeps coming back, you know, relentlessly, as this kind of phoenix, you know, you destroy him and he comes back. But obviously it's a different one, right? So you have to buy into this conceit that there are a number of identical, Arnold-based Terminator T-800s from the Skynet future, that are sent back through time to do various missions. And to see how they each turn out differently, to me, was kind of intriguing. So one is programmed to kill; remorseless, relentless -- this was the first guy, obviously -- no conscience, nothing. The blankness, the blankness of his affect; completely inhuman. Even though he's meant to be an infiltrator, which is the big joke of the whole thing, because Arnold doesn't exactly blend, you know. Then you've got the one that's designed, that's programmed, for good. He's not innately good. He's programmed to be good. He doesn't even really want to be good; he'd rather kill people, 'cause it's simpler. And he's told not to, and he kind of resists it, and that's kind of, there's a little, kind of, ironic dark humour in movie two, around that idea. But what happens if you leave a Terminator, with his learning computer, with his neural-network CPU, just, with no mission, and no purpose, awash in human society for two-and-a-half decades? What happens to that guy? To me, that was interesting. You know, where does he wind up? His job is to try to study, try to mimic human behaviour, to the extent possible, to parrot the way people speak, to learn. We saw it right in the first film, you know; the guy's yelling at him through the door of the hotel room, you know, and he's got a number of answers, and he chooses one, and he says it, and it works, the guy goes away, you know, we see him learning. It's kind of an intriguing idea. So I thought, "Let's follow that guy."

_{James Cameron Interview - Terminator: Dark Fate}