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.
Sarah's realisation that the T-800 is actually the best possible father John could have.
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.