Sixth movie: Same production as the 5th film, but with the help of James Cameron's Lightstorm Entertainment (Paramount distribution in the US, Fox/Disney worldwide)
This also led to the first two being released in home video by a plethora of companies. The first film's distribution ended up being the wildest, going from HBO to Hemdale to Live (under license from Epic) to MGM.
James Cameron was contractually obligated to give T2 to Orion to option, but the filmmaker's asking price was so high that the studio had no choice but to pass. It is a fact that Cameron, Schwarzenegger, and producer Gale Anne Hurd despised Orion's treatment on the first film (although it was a hit), that the high asking price has been perceived to have been deliberate so they wouldn't have to work with the studio again. If the situation was different, Orion might have released both films.
Of a very scary sort. The British Ministry of Defence actually operates a satellite network used to coordinate unmanned vehicles - including "Hunter Killer drones" - called SkyNet.
The US Air Force has a unit readiness tracking system called, I shit you not, SkyNet. During exercises, announcements come over the loudspeakers for group commanders to "update numbers in SkyNet".
There is a company called Cyberdyne that is working on exoskeletons. Based in Japan. The version it's getting the most attention for is called the HAL 5. Also, none of their projects involve AI. Although the Cyberdyne name isn't meant to be a reference, as they work in Cybernetics, and dyne is a suffix meaning power. Doesn't explain the HAL 5's name, though.
HAL is an acronym. It stands for "Hybrid Assistive Limb".
If parts of T4 gave you nightmares then just skip this link. At least there's no... teeth.
News 4 in Tucson, Arizona has a weather, traffic, and safety observation network called Skynet:  They have billboards for it all over the city and it's a bit unsettling. (Especially when in The Sarah Connor Chronicles the Los Angles traffic system was originally destined to be the "nervous system" of Skynet.)
Franchise Zombie: Although he would give some praise to the sequels here and there (mostly because he's friends with some people involved including actors), James Cameron wanted the franchise to end with Judgment Day. None of the myriad of producers who came to own the franchise thought much of this and sequels were produced. At least two of them, Salvation and Genisys, ended up stillborn franchises with diminishing returns in both critical reception and box office. The producers of Genisys had another go at it with Dark Fate, but again, Cameron's real involvement in it doesn't amount to much.
God Does Not Own This World: The rights at first were equally shared between James Cameron and the first film's co-writer/producer Gale Anne Hurd. By the second movie, Cameron's part was with Carolco, which went belly-up. The two producers who formed Carolco (and went on to form C2) purchased the Cameron share in the company's liquidation auction in 1998 and Hurd's share one year later. To prevent the eventual subversion of the trope, as once the first movie turned 35 the rights would revert to Cameron, the Genisys producers who currently own the franchise went to get his help for the sixth movie (although he only produces and helped create the story, as the Avatar sequels kept him very busy). However, Gale Anne Hurds filing of a copyright termination means the rights could potentially return to her and Cameron.
Older Than They Think: The line "Hasta la vista, baby" was used by Bob Hope in 1970, Jody Watley's "Looking for Love" in 1987, and Tone Lōc's 1988 single "Wild Thing," the latter's music video inspiring Cameron to include the line in the Terminator 2 script.
John Connor has been portrayed by numerous actors; with 7 actors playing him in 4 movies and a TV show. Sarah Connor was recast for the T:SCC series as well, with Lena Heady taking the iconic role from Linda Hamilton and ironically sharing the same initials.
T4 also has a new Kate Brewster and Kyle Reese; understandable and necessary in the latter case, since he's several years younger than in the original film.
Then came Genisys, which recasts Kyle, John and Sarah.
The Production Curse: More legally cursed than anything. See the Channel Hop trope above? That occurred because the production companies (and a couple of the distributors) kept folding before the next sequel could be made (only Hemdale was lucky enough to be hanging around when its film's sequel was made, and even then they had no involvement in T2 due to selling the IP rights to Carolco to settle ownership disputes).
Sequel Gap: 7, 12, 6, 6 and another 4 years between installments.
Word of Saint Paul: The novelizations of the first two films were written by a close personal friend of James Cameron, who had also contributed rough ideas to the concept. Some of the details revealed in the novels are confirmed true in later films, in creator interviews, or in other works—for example, the series of Terminator that is after Sarah. It was only called a T-800 in Terminator Genisys (the fifth movie, 21 years after the original!), but the novel refers to it as a Series 800.
Interestingly, the three films themselves work as a loose trilogy, with the third and fourth movies continuing the story from the first two films, and then the fifth forming an alternate timeline and returning to the past.