[Delvo]

There's also the physical nature of the technology/network that created the disaster, its development/creation, and what its actions were leading up to the war.

T2 said it started with the creation of a new type of CPU by Miles Bennett Dyson, which was to be used in individual buildings/compounds and to run independent machines like airplanes and tanks. Only after doing so "with a perfect operational record" for some time did such machines inspire Congress to pass a bill to use those multiple independent CPUs as the basis of a network. And that was done according to plan because they did the job better than humans. That network's development of sentience was an accident resulting from the advanced hardware of the CPUs that had been networked.

T3 accepts that that all got stopped and that the CPU wasn't invented at all; it has the military network "Skynet" (different thing, same name) as the first step in the story, created as a "dumb" network from the start without anything special about the separate parts within it, using regular modern computers, instead of being a derivative of Dyson's CPU because that didn't exist. Lacking that advanced hardware and its inherent ability to become sentient by itself, T3's "Skynet" was only rendered sentient by something from the outside: the TX's invasive software and nanobots. And in T3 the network was activated by humans in a moment of sudden desperation when all else seemed to be failing (due to the TX's interference), not on a schedule as planned while things were going fine as in T2.

There might also be a difference in how the two different "Skynet"s got the idea that humans are the enemy. In T2, it was self-defense because humans tried to turn it off when they realized it was more than they'd bargained for. I don't recall how it came to that conclusion in T3, but it doesn't seem like it could be self-defense; did it actually come from the TX?