How can the Resistance shut down Skynet in the future? According to the voiceover at the end of the Rise of the Machines:
There was no system core; it could not be shutdown.
and yet in Terminator Salvation, the Resistance seeks to destroy
Skynet's central core
Is this a plot hole, or another timeline alteration?
It's an alternative timeline. These dates are taken from Terminator wikia's timeline of events:
Note that the dates given are prior to the events of the movie. T-2 caused J-Day to shift to 2004 or 2011 (depending on which continuity you want to go by). Salvation is the odd one out that doesn't seem to fit anywhere, except that it's supposed to be a prequel to T-1, in which the date of J-Day was apparently never stated.
It could be that the date from Salvation is the original timeline, and the arm left behind in T-1 caused the date of J-Day to shift from 2003 to 1997 - which would mean that every single depiction of the future is from a different timeline, and could explain that discrepancy.
According to the Terminator wiki
Skynet's Central Core was located deep underground within Skynet's main complex. From intercepted surveillance of video data feed, the Resistance was able to hypothesize that this Central Core was a form of cold fusion reactor, needed to supply Skynet with its tremendous energy requirements. This Central Core is the key to Skynet's ability to operate; Skynet would be rendered virtually disabled without it. It has therefore become the main target for the most skilled Resistance strike teams. The Central Core is protected by the T-1000000, and thus there has, as of yet, been no successful strikes against it.
So, even though Skynet could not be shutdown, its power source can be knocked out.
Skynet initially existed on all internet connected devices, right up until the moment the bombs fell. The nuclear strike would have wiped out many of the computers in offices, dorm rooms, and houses, dropping power grids, and possibly even severing lines of communication for the internet itself leaving SkyNet itself weakened much like a stroke victim. This could also explain why most of the action takes place in California, it may have one of the more robust internet infrastructures designed to survive nuclear war as well as earthquakes.
A Central core may have been all that was left after the initial nuclear strike. From a learning point of view, Skynet saw that being everywhere was dangerous because shortly after it existed everywhere, most of it died off.
Personally I put this down entirely to Terminator franchising. If you can make money with a fourth or even a fifth film, then why not?
I do not think the audience are supposed to read into this too much.
The time between T3 and T4 would probably be quite a while due to nuclear war etc. So it's somewhat open to interpretation. One might say that "Skynet was just software in Cyberspace" - OK, fine, but nuclear attacks fry electrical equipment rendering it useless, so how can Skynet sustain itself against its own attacks, when the majority of it's existence is spread over "cyberspace"? Skip to T4, Skynet now has a central core. One might interpret this as the last remaining stronghold of Skynet. Why? because Skynet has destroyed itself elsewhere, therefore to compensate, it moves all of its remaining resources, knowledge etc to one central location, and makes it a fortress which is difficult to enter (unless you are a machine).
This might also be attributed to the change of directors over the story. We all know the best films were Cameron's, however AFAIK, T3 was the Windows Millennium Edition of Terminator. It really was a flop. I mean, Arnie wearing Elton John glasses....WTF!? Terminator 4 was somewhat redeeming, and it was nice to see the story enter the nuclear war as its a fundamental part of the story.
