Using the definition provided at the end of your answer, I think it can be argued that both versions of Aliens are director's cuts, in that both versions appear to have received Cameron's stamp of approval.
However, judging by various remarks made on the commentary track for the Special Edition (as well as the intro you quoted), it sounds like that's his preferred version of the film, and that he only agreed to make cuts after being persuaded to do so by others, with the primary aim of upping the film's box office gross, rather than because those cuts made the film better in his mind.
The film's producer, Gale Anne Hurd, stated the following on the commentary track, at about ten minutes into the film:
"This scene was one of the seminal scenes in the movie, and was one of the ones that had to be deleted and omitted from the theatrical version because of length. We didn't have multiplexes back then, and only so many showings a day that you could have of a film, and we had to get it no more than two hours and ten minutes in order to get the maximum number of screenings per day."
On my DVD copy (from the nine-disc Alien Quadrilogy box set), the runtime for the theatrical release comes in at precisely two hours, 11 minutes and 34 seconds, putting it just over the required length stated by Hurd, but I'm guessing that was close enough for it to receive the maximum number of daily screenings she spoke of. And that seems to be the main reason why the theatrical release was cut to that specific length.
To shed some more light on Cameron's perspective, he had this to say on the commentary track, at about 14 minutes into the film:
"The tag of this scene is gonna be a throw to this big sequence that takes place on the colony, which is before the aliens attack, and that's cut out of the release version. So this, coming up, is the biggest single change from the release version of the film. And it's an entire reel. And the funny thing was, I'll never forget, Gale Hurd, who was my wife and my producer at the time, we were confronted with this issue of how do we reduce the running time of the film by close to 20 minutes, and I just couldn't see it, I just could not see how it was possible to do it. A cut here and a cut there, a few seconds, bit of a scene, the tag of a scene, maybe. She said 'I've been thinking about this for a few days.' And I said 'Alright, well go ahead.' She said 'Reel three.' Which starts right here. 'You can take out reel three.' And of course I immediately rejected that as completely absurd. And then I thought about it, and reel three begins here and it ends with Newt's scream, when her father has the Facehugger on his face. It works flawlessly. It's a brilliant cut, and I have to credit Gale with that. Now I had poured a lot of energy into the design of this, of these scenes, and the alien derelict ship. I think the biggest problem for me was that I couldn't imagine this film without the cognitive tether to the first film of the alien derelict. But it turns out that it works perfectly. A little dialogue bridge, and it works fine."
Note in particular that part about them needing to reduce the runtime by close to 20 minutes. This implies that something resembling the cut we now know as the Special Edition was actually the original cut of the film, and that the theatrical cut was made afterwards, not before.
Cameron also said this, at about an hour and 26 minutes into the film:
"Okay, so this is an added scene with these sentry guns. This is a scene that got the axe as a result of the studio's idea that we were wasting too much time not really getting on with the story. I actually think this stuff really ups the ante, and increases the fear a lot."
And he said this, at about an hour and 37 minutes into the film:
"I think editorially it's a fun moment. I like the build. I didn't do any editing directly on this film. But, I think that especially because it's film-based editing, with that kind of rapid cutting, I thought it had really good energy. You see alot of that type of cutting now with the Avid obviously, 'cause you don't have to make every splice manually. This is the Alamo told with six people. So the sentry guns are just the equivalent of the first couple of attacks. You know, I think it's great foreboding. It's just coming at them like a wave. If these guns don't stop them, they're screwed. I actually think it works pretty well, but the studio talked me into taking it out. So I'm happy to see it restored in this extended version."
There's no suggestion that Cameron was forced into making cuts he didn't ultimately get on board with, but these statements do indicate that he was persuaded of the merit of the cuts, rather than being the person who proposed them in the first place.
The block quote just above also clarifies that Cameron didn't personally make any edits to the film. Ray Lovejoy is credited as the film's editor, although Cameron indicates on the commentary track (at about one hour and 24 minutes into the runtime) that Lovejoy's edits were submitted to him for his approval:
"I hired Ray for one simple reason: 'cause he had worked with Stanley Kubrick. It took him a while to really get what I was trying to do with this movie. And a lot of his early cuts, quite honestly I didn't really care for. And Ray was getting really frustrated. And I remember toward the end he cut the Alien Queen battle, you know the Power Loader Queen battle at the end of the film. And he was really nervous, because he hadn't really given me the kind of action cutting that I really wanted, and I'd had to mess with it a lot, and there was another editor who was cutting some stuff, and I was liking his action cutting better than Ray's. And Ray is just a dear guy and a really good editor, but he was struggling with it. And so, finally, he just grabbed all the film, locked himself in his room, said 'Don't bother me.' Not mean or anything, but 'I just gotta do this.' And he went in and he cut the entire last eight minutes of the picture. And he showed it to me very nervously -- he cut it in like a day or two -- and he showed it to me very nervously. And I watched the whole thing, and I said 'It's perfect. It's absolutely perfect. Don't change anything.' And that was that. And he felt like it was such a huge victory, because he had actually got it. He had mastered the style for the film."
Finally, we should consider the fact that Cameron not only directed Aliens, but wrote the screenplay too. Therefore, all the scenes which were removed from the theatrical release, and later put back into the Special Edition, were scenes he personally scripted, and decided were worth filming.
So, if you define a director's cut simply as a cut that is approved of by the director, then it seems both versions of Aliens can be considered director's cuts. However, all signs point to the Special Edition as the truer reflection of Cameron's personal vision for the film.