So we've seen it work the way you're describing it - you go to the past, change something, go back to the present, the present is now different.

But I don't think we've seen it from the perspective of someone not going to the past. E.g., you go back to the past, change something, I stay behind in the present - does history change for me instantaneously? Obviously not, since we have the example in the original post here. So is there some delay, where the changes haven't "caught up" to me yet to alter my present (like Back to the Future)? Or does it never catch up (like in the example where changes move forward at 1 second per second, time traveling forward may put you into a changed future, but an observer left behind may just remain in a present that never gets hit by those changes)?

My point isn't to try to claim that I know how it should work - my point is just that people tend to criticize time travel plot holes when the story doesn't even specify how time travel works exactly. There are a lot of different ways it /could/ work, and unless the story lays it out exactly, then making up your own head canon for how it works and then claiming the story has a plot hole because it didn't work that way, is just silly.