I think you can effectively sort post-apocalyptic settings along three different axes, perhaps with some extra free-floating modifiers to indicate unusual twists. Axis 1 is speed:
- Did the end come faster than anyone could react (instant apocalypse)?
- Did it happen within one lifetime (fast apocalypse)?
- Was it drawn out over more than one generation (slow apocalypse)?
Axis 2 is recency:
- Can almost everyone remember pre-apocalyptic civilization (near apocalypse)?
- Is almost everyone too young to remember the Beforetime directly (far apocalypse)?
- Has even the fact that there was an apocalypse become forgotten (remote apocalypse)?
Previously, I labeled recency from the view of an apocalyptic historian, calling them "early" and "late apocalypse". But an apocalyptic setting is really about how the loss of civilization affects the characters immersed in it, so maybe "near" and "far" are better terms. Remote Apocalypse needs to be distinguished because, in some cases, a new civilization is in place, so at best, the discovery of information about the pre-apocalyptic civilization sheds light on the new civilization, but does not involve a sense of loss.
Axis 3 is severity:
- Was everything destroyed (total apocalypse)?
- Was it just a standard breakdown?
- Was it just a partial breakdown, forcing severe changes (semi-apocalypse)?
You can further modify the severity by making it regional. Dark Angel, for example, involved an EMP taking out the banking system in the US and causing significant breakdown on the West Coast, but the rest of the world only suffered slight setbacks as an indirect result, which makes that setting a regional instant near semi-apocalypse. I used to completely separate situations like this from an apocalypse, calling them "cataclysms" instead.
A side note: In a total apocalypse, you have to import outsiders to experience the aftermath. These may be off-world archaeologists or a new race that evolved to replace the one that was lost, or time travelers of one sort or another. You really don't have much of a sense of loss except with the time travelers.
Another modifier is "mid", marking an apocalypse as still being in progress. Almost all slow apocalypses are presented mid-apocalypse, but "Panic in Year Zero" is an example of a fast mid-apocalypse situation. Many mid-apocalyptic settings ditch any deep thoughts about loss of civilization and focus purely on survival.