Combined with one of the biggest Tear Jerker moments ever, John's breakdown when he realizes that the T-800 has to destroy himself to prevent the creation of Skynet. The kid has been through hell the entire movie, and its here that he completely loses it at the thought of losing not just his bodyguard, not just his friend, but the closest thing he has ever had to a father.
According to James Cameron himself, this one moment was the axis around which the entire movie turned:
"On T2, I wondered if I could get the audience to an emotional place where they would cry for the Terminator. That was my goal: Could I take world's coldest motherfucker and turn you around in a two-hour time period to where you actually felt sorry for him? Forget about all the hoo-ha with the liquid metal guy: that was fun, but getting the audience to cry for the Terminator was the big cinematic challenge. That's the reason I made the movie."
One scene stands out in T2: "Hey wait, you swore!" "Trust me."
Bonus points for the T-800 finally getting the smile right.
And when he drives away the police with a gatling gun and a grenade launcher, the T-800 scans the area. The screen from his view reads "Human Casualties: 0.0". He did learn, despite of the tremendous amount of Loophole Abuse he pulled off.
"I need a vacation". The T-800 actualy learned how to humor!
Unfortunately, recognizing a Terminator's worth is very likely what drives her to take that shot at killing Miles Dyson. She was obviously thinking about it when the T-800 shared his inside knowledge of Skynet's rise to power, but put it off because they were already safely halfway to Mexico and she felt her first duty was protecting John; even if she pulled it off, heading back to LA to kill Dyson would almost certainly result in her either being killed by the T-1000 or arrested by the cops followed shortly by being killed by the T-1000. Once she believed that the T-800 was more than capable of taking her place as John's protector and mentor, she liked those odds. She entrusted John to the T-800 for what may be the entire rest of his life.
When John tells the Terminator to reveal himself to the Dysons, he takes their son Danny away to see his room and spare him more trauma. It's especially sweet when you consider that John didn't have much of a childhood himself and would rather not have an innocent child caught up in this any further.
The T-800's backup power activating becomes one in the special edition, thanks to the scene of Sarah and John removing his block on learning and developing. This makes it clear that what's actually happening here is that he's no longer just following his programming, but actually cares about John and Sarah and is going to do whatever he can to save them.
More specifically, he goes to help Sarah. His mission is to protect John, so him helping Sarah was a conscious decision even though she doesn't like him at all.
The scene where John laments to The Terminator about his life up to meeting him. It's pretty heartwarming that he can confide to a machine who at the time didn't understand human emotion, but was willing enough to listen.
In a blink and miss it moment the reunion of Sarah and Enrique has their dog not actively barking or snarling at the Terminator. This isn't a case of forgetting as Max John's pet dog is very much freaking out to the point John notices it when the T-1000 is impersonating Janelle. It's clear that Enrique's dog doesn't find the Terminator a threat, and in fact shows very subtly that the Terminator is gradually learning about the value of human life.
The alternative ending that remained in the deleted scenes. We see an aged Sarah Connor in the playground of the future, but not destroyed by nuclear war. She monologues hope was given to her by the Terminator. An adult John Connor, who is, in stark contrast to the opening of the film - he's not heavily scarred or grimly waging war on the machines, but happily playing with his daughter. A future given to him by the Terminator.