An hour long discussion with the man about time travel.
I've always gone with the multiple timelines theory, but life_on_queen argues there's a single timeline in the TSCC verse. So after a brief "single timeline!" / "multiple timelines!" exchange, I took the discussion to real life and mentioned it to the man. He wasn't interested in talking about it. "Single timeline," he said. "Let's watch TV."
I grabbed pen and paper, started drawing diagrams and talking about Derek and Jesse. Next thing you know, the man is all into it, grabbing my pen and paper and drawing his own diagrams. Much debate and pages of back and forth scribbles later (with a very annoying pen that kept refusing to write), he declared there must be multiple timelines (and also "let's watch TV").
Here is a coherent version (I think) of what we came up with:
I got to pondering when exactly a new timeline is created. When my husband and I were discussing it, we went with a new timeline being created as soon as someone arrives. Really, we both agreed that the new timeline wouldn't be created until the time traveler actually did something to affect the future. But without knowing how significant such an event has to be, we went with the butterfly effect/chaos theory — that the slightest alteration of anything could result in a chain reaction that leads to a different future. Therefore, a new timeline is most likely created as soon as someone arrives (especially since time travel in the terminator world involves electrical disturbances, big holes cut into the landing spot and naked time jumpers walking about).
The problem with that is it doesn't mesh with T1. In that, the terminator and Kyle did not arrive simultaneously. The terminator arrived first and, by this theory, would have triggered a new timeline. If that had happened, though, Kyle would not have been able to intercept him and Sarah likely would have been killed. Well, that's no good.
For the sake of the film, it looks like I'm going to have to skip the chaos theory and wait for something bigger to trigger a new timeline. Of course, the terminator kills some punks very soon after arrival in T1 and that's clearly action that would impact the future. But I'll have to wash that away by suggesting their deaths are irrelevant until other people are affected by them or until the punks missed doing something they would have done had they not been killed. Since they were out partying, it's possible they would have passed out and not been missed at all until the next day when Kyle had already arrived. It's very thin logic, but then again, we're dealing in the theoretics of time travel as applied to a 24-year-old work of fiction :P
Soooooo... in the terminator universe, a new timeline is not created until something significant has happened (as a result of the time traveler's presence) that causes a chain reaction of events that alters the future's previous path. That doesn't necessarily affect my diagram since I'm dealing in years and not specific dates. But it does mean that it's possible for a time traveler's original future to remain intact for a period of time, allowing someone else from that same timeline to follow them. It also opens up the possibility that not every time jumper actually creates a new timeline.
To put that into a specific TSCC example, it could be that the discovery of Andy Goode's death was the thing that finally caused a new timeline to emerge from Derek's time jump. Or it could be that Vick actually came from the same timeline as Derek and it was his killing of Barbara that caused a new timeline. Whichever it was, that would also be the point, then, that Derek's original future was closed off (people from his future could no longer get to the timeline he's in now). And that brings us to Jesse. She and Derek clearly share some memories, but they also have different memories because the future was altered. Well, if her future was different then she could not have come from the same timeline that he did. She came from the new timeline created when Andy Goode was found dead (or Barbara or whatever event it was that created the new future).
This Jesse never actually knew our Derek. And our Derek never knew her. She knew the young Derek who's now just a kid in the current 2007 timeline. It's technically the same Derek, but since his future is changed to some extent, he won't end up the exact same person as our Derek — he won't ever know Billy Wisher, for example. Likewise, Jesse's not the exact same person our Derek once knew either. This Jesse is also just a kid someplace in 2007 who will grow up in a future that's different from the one in which she knew our Derek. Make sense? Heh. Probably not. Don't worry. You don't need to understand it. I've got it all worked out right up here *taps head* ;)
One last thing I'll throw in here is that I don't really see timelines as being these rigid lines fanning out away from each other. I think they're more like spiraling paths that intertwine and can even intersect. If Derek and Jesse share identical memories of any specific event, for example, that could be a segment where their different timelines actually overlapped. Of course, that also means it would theoretically be possible for someone to jump from one timeline to another. But in order to do that, they'd have to actually know where and when those timelines intersect and I can't think how that would be possible.