> There likely won't be one Skynet, but rather multiple AI's, produced by various sponsors, starting out as relatively harmless autonomous agents in corporate competition with each other

This is kind of how Skynet begins in the TV series Terminator: The Sarah Connor Chronicles. It takes place after the 2nd movie in an alternate timeline from the 3rd movie (and establishes itself how that's possible without contradiction).

Specific examples I remember: A chess machine for the brain, traffic light cameras for the eyes and ears, a repurposed factory to build the terminators themselves. The series is about the Connors getting information from the future and going on the offensive to prevent Skynet from forming.