Now we all know the dates/times/ages stated are a little off and don't line up in the movies however for the sake of arguement let's just put it aside for a moment. (like in T-2 the computer that the T-1000 reads in the police car at the beginning of the movie shows that John is 10 years old and was born in February of 1985. This means that the movie needs to have taken place in 1995 (2 years before Judgment Day). Yet Arnold claims in the car, while driving in the desert, that Judgment Day would happen in 3 years, just after the Skynet funding bill is passed by congress. This comment would make it 1994. Little goof.)
Let's look at the Cannon Material, Timelines, and Mind Benders:
In T-1 Reese says to Sarah while in a parking garage, hiding in a second get-away car. "....defense network computers. New. Powerful. Hooked into everything. Trusted to run it all. They say it got smart. A new order of intelligence. Then it saw all people as a threat, not just the ones that were on the other side. It decided our fate in a micro-second. Extermination."
Note that we have to realize the source this is coming from. Reese is a soldier from the future who states he didn't see the war start because he was born after Judgment Day. All his knowledge is second-hand and possibly not the whole picture.
The entire time line from T-1 to T-3 is a single time-line which I would call Time-line ONE. Now before I jumped all over, what I mean by this my thoughts on how the writers could explain this in a story-like way. Real quantum theory (and the sub-topic of time travel theory) wouldn't work the way the movies try to exhibit it, but these are movies and a story told for drama, entertainment and action...not to get the science exact so as someone who enjoys Quantum Theory and the Terminator series, I can put on my hat of" suspended disbelief".
Time-line ONE begins with the late 1990's/early 2000's when Skynet is built and Judgment Day occurs. The battles wages between humans and machines. In 2029, the first Terminator and Reese are sent back in time to 1984 to kill/protect Sarah Conner respectively. Reese sleeps with Sarah and in doing so, is shown to be John Conner's father. Nothing new that we didn't already know but I'm just getting the basic facts down. Following this time-line, when the year 2029 is reached Reese will always go back and create John.
Anyway, some would say, "WAIT! That isn't right!" and point to T-2. Dyson, the arm and CPU of the first Terminator, and Dyson's statement that it was radical stuff that took them in directions they would have never thought of. Just hold on to this thought and I'll get back to it in a moment.
In T-2, the scene just after the gas station in the desert, Sarah asks how Skynet gets built. The T-800 Model 101 (Arnold) is driving the station wagon. He explains how Dyson builds the neural net processor and Cyberdyne becomes the military's biggest supplier of computer systems.
T-800: "...The system goes online on August 4, 1997. Human decisions are removed from strategic defense. Skynet begins to learn at a geometric rate. It becomes self-aware at 2:14am Eastern time August 29th. In a panic, they try to pull the plug."
Sarah: "Skynet fights back"
T-800: "Yes. It launches it's missiles against the targets in Russia."
John: "Why attack Russia? Aren't they our friends now?"
T-800: "Because Skynet knows the Russian counter-attack will eliminate its enemies over here."
If we step back it is possible to see that both explanations on Skynet; the one from Reese from T-1 and the one from the T-800 from T-2; are intertwined up to this point in the T-2 movie.
Let us focus on Reese's statement and the T-800's statement.
1. Both agree Skynet is hooked into all military equipment, specifically strategic defense and is trusted to handle all the decision making (actual point for all Terminator related material).
2. Both agree Skynet evolves and becomes self-aware. Reese implies it via "a new order of intelligence" and the T-800 comes right out and says Skynet becomes self-aware (true Artificial Intelligence). Same difference.
3. Now Reese says "Then it saw all people as a threat, not just the ones that were on the other side" while Arnold states "In a panic, they try to pull the plug" and Sarah finishes the thought "Skynet fights back". We can see how both statements could be the same, just from different viewpoints. Skynet becomes self-aware and the military/scientist try to cut it's control/shut it down/kill Skynet. Remember, Skynet was primarily programmed to protect America and view the Russians (or any country that isn't America for that point if we are thinking in a U.S. Military mindset) as a threat. When it's creators, the 'American good guys' which it was designed to protect try to kill it, Skynet sees all people as a threat, not just the Russians. Skynet decides to protect itself by killing off or (as Reese said) beginning the "Extermination" of those who sought to shut it off.
Of course, now Skynet can't just sit back and sigh in relief because those pesky scientists and military men who tried to kill it are gone in a nuclear fire. First, the Russians are going to be pissed for obvious reasons and will want to kill off this Artificial Intelligence (Skynet) who attacked their motherland. Heck, going one step further mankind as a whole isn't going to sit by and just say "well OK, you have the right to exist after you just tried to commit genocide on our entire race". Mankind is going to want retribution and Skynet knows this. The war begins.
So now because of interference in the time-line (back in T1) by future Skynet, Time-line TWO is created where Cyberdyne has the first Terminator's CPU and Arm and is set to create Skynet far sooner than in Time-line ONE. Note that in BOTH time-lines, Kyle Reese goes back and creates John so that is not going to change, only the date that Judgment Day occurs and why/who creates Skynet. Time-line TWO, again future Skynet is threatened by the resistance and sends back the T-1000 to kill John. John sends back another T-800 to protect himself. In doing so, Skynet inadvertently steps on its own feet. If Skynet didn't send back the T-1000, there would be no need to send back the T-800. Sarah and John quite possibly would never have learned of Dyson and the new, 'sooner' Judgment Day would have happened. Of course an argument could be made that considering how it was implied through veiled comments made by John in T-2 to his friend Zack that Sarah seemed to watch Cyberdyne like a hawk and was caught trying to blow up one of their facilities (he just states a computer facility but she seems fairly focused on Cyberdyne the entire movie so the fair guess says it was a Cyberdyne lab). This is what lead to her arrest and placement in the mental institution. This is just speculation though.
Back at hand, once Dyson is killed, his research destroyed, all Terminator parts (First T-800's CPU and Arm, Second T-800, and T-1000) are destroyed. [Ok, nitpickers will point out that the second T-800 lost it's arm in the steel mill fight and that leaves evidence of the Terminators. Real nitpickers will point out that Cyberdyne may have only given the Arm and CPU to Dyson while the the leg that was damaged while the T-800 was being dragged under the semi in an earlier scene (and shown for a good 3-5 seconds right after the pipe bomb goes off inside the T-800, and not looking any worse than it did attached to the T-800) might still be around in another lab.] Judgment day is now reset back to it's original date. Some could say that Time-line TWO now is changed to Time-line THREE, an unknown future as portrayed by the headlights on a roadway at night, however I would like to think that Time-line TWO reverts back to Time-line ONE. They are one and the same. The time-line changes so Judgment Day is pushed back further into the future possibly back to it's original date from the first movie. So when Kyle Reese told Sarah Conner in the first movie about Skynet, the truth to his words had come back full circle...back to the original Time-line. Basically, Judgment Day was always going to be the date it was originally stated at from the first movie. Kyle just didn't know about Dyson or much about Cyberdyne having any tech from the future to work with.
As for Skynet winning, yes it starts the war with a winning move and nearly wipes out humanity. If you go back and watch the first movie however, Reese states while he is in the police station being questioned:
Doctor Silverman: Why didn't the computer just kill Connor then? Why this elaborate scheme with the Terminator?
Reese: It had no choice. Their defense grid was smashed. We'd won. Taking out Connor then would make no difference. Skynet had to wipe out his entire existence.
Doctor Silverman: Is that when you captured the lab complex and found... What is it called? The time displacement equipment?
Reese: That's right. The Terminator had already gone through. Connor sent me to intercept and they blew the whole place.
In T-2, Doctor Silverman puts a date to when Reese jumps back, 2029. So by 2029 Skynet is defeated and the humans have all but won the war. It was Skynet's last ditch effort to survive. Note that Reese doesn't know for a fact the facility is destroyed. He can't. He was sent through before it was destroyed so what happened to it afterward he can only assume. If it was destroyed is anyone's guess.
A side-note here: I think a subtle point to all the Terminator movies is the fact that no matter what was done, humanity was/is always going to be destined for the Technological Singularity (computers being smart enough to evolve and create better versions of themselves without human involvement.) and that Time Travel is dangerous and not to be tinkered with.