The "Extended Cut" is an innoficial version which has been released the first time on the Ultimate Edition DVD from Artisan. You have to put in the date of the Judgment Day (82997) to unlock this Version of the movie. However: It is not a Director's Cut or whatever.
The T-1000 re-enters the house. He walks into John's room, his hands touching John's belongings to get information. He stops at one poster on the wall, then tears the poster away. Hidden behind it there is a cardboard box saying "mum". The T-1000 opens it up. Inside there are photographs and letters from Sarah. This is when the T-1000 finds out that Sarah is in the sanatorium in Pescadero. He therefore decides to go there. (remark: In the first movie Kyle tells that most of the data was lost in the war, only the name Sarah Connor and the place Los Angeles were known.)
Here is a difference between the Special Edition and the Ultimate Cut:
Special Edition: We see the street. The voice of Sarah Connor says: "The unknown future rolls toward us. I face it for the first time with a sense of hope, because if a machine, a Terminator, can learn the value of human life, maybe we can, too."
Ultimate Cut: We see Sarah Connor in the year 2027. She is sitting on a bench on a playground, recording an audiotape: "August 29th, 1997 came and went. Nothing much happened. Michael Jackson turned forty. There was no "judgment day." People went to work as they always do. Laughed. Complained. Watched TV. Made love. I wanted to run through the street yelling, to grab them all and say, "Every day from this day on is a gift. Use it well." Instead, I got drunk. That was thirty years ago. But the dark future which never came still exists for me. And it always will-like the traces of a dream. John fights the war differently than it was foretold. Here, on the battlefield of the Senate, his weapons are common sense and hope."
Sarah's granddaughter approaches, yelling: "Tie me, gramma! Tie me!"
Sarah: "How's that?"
Girl: "Thank you, gramma."
The girl runs back to her father, John.
Sarah: "The luxury of hope was given to me by the Terminator. Because if a machine can learn the value of human life maybe we can, too."