All the times I've watched this movie and I never caught the fact that his name isn't John Reese!
From Sarah's view at the time, it could be that John's father and her were together for a time but never married and she had to raise John on her own and the father exited their relationship when she was carrying John or after he was born, but then John is the result of a never ending Paradox loop, if you take T3 into account, no matter what Jugement day was always going to happen and Kyle would come to exist and one day, go back in time to save Sarah, without knowing he is John's father, John is told little of his father and he takes steps to ensure that stays alive and Kyle goes back in time to save his mother and it just keeps going and Skynet cannot learn because its trapped in the loop as well, as its role in the loop is to be the threat that has Kyle go back in time and the two protector Terminators to also go back and protect John, once as a boy and then as a young man, living off the grid, it never ends, not until John is forced to kill Kyle in the future before he goes back in time, thus erasing himself in the process and completely changing the course of history, but John doesn't want to do that as he wants to live!
The original Terminator movie is my favourte, as it builds the suspense up carefully and nicely and to start with, you not sure exactly what is going on with these guys appearing Naked in back allyways, its only after Kyle saves Sarah that we start getting the information drops about the future war and Skynet trying to erase its arch enemy before they were born, Sarah was weak and vunrable to start with, but she finds strength and courage and finally puts an end to the threat to her life, yet it comes at a cost! T2, while it builds upon the first time, its not as gripping as the first one as that was made on a smaller budget and there isn't the same character growth as you get in the first film