post and comments.
In the two temporal dimensions scenario, the original history is "snuffed out" from the point of view of a hypothetical observer for whom the first temporal dimension is a fourth spatial dimension but not from the point of view of any inhabitant of the original history. The fact that a particular individual did live in part of the original history is not negated or "snuffed out" either by the fact that that history also contains periods subsequent to his death or by the fact that the entire history has, from the point of view of the hypothetical observer, receded into the past of a second temporal dimension.
If it is said that the original history still exists from the point of view of the inhabitants of the subsequent history, this implies that (i) the two histories coexist in a fourth spatial dimension rather than that (ii) the subsequent history has succeeded the original history along a second temporal dimension. Travel between histories is more likely in (i) than (ii). Time Patrollers can travel back and forth within the current timeline but not to earlier or later timelines - although I speculate about one way to reach earlier timelines.
If a large temporal vehicle with many passengers entered the "deleted" Carthaginian timeline, then that vehicle still exists until its departure point in the current timeline. If someone now time travels to before that departure point and stows away in the vehicle, does he enter the "deleted" timeline with it or does his presence in the vehicle alter its quantum status so that it now moves futureward along the current timeline?