The method I use to determine if a new level is an aquifer is to check what material it's made of. I'm getting fairly good at remembering which materials are capable of containing an aquifer and which aren't, but the wiki is a good place to check. There is a recommendation to check whether it's damp or not on the exact tick the tile is dug out, but I find checking the material to be a better method.
If there are other materials inside an aquifer carrying layer (such as e.g. gem clusters), those are usually not aquifer carrying as gchristopher pointed out, and thus won't drain water. Puddingstone is an exception, as it's an aquifer carrying material within an aquifer carrying layer (I'm not sure if there are others).
As indicated, gems can throw a wrench into the machinery, and very occasionally it's possible to dig completely inside a gem cluster (but typically for a single level only).
A DF "save" is the RegionX folder within the Save folder, not the Save folder itself, as that typically contains multiple saves, plus the working Current folder.
Looking at the second save set (region2), I see a normal two level deep aquifer, not 3. Above it you have Clay Loam (too high to up to contain an aquifer, and clay never has one anyway). The aquifer is first in Yellow Sand and then Loamy Sand, and below that you have Dolomite, which isn't aquifer bearing.
Aquifers start at the 3:rd level from the top, never above that, and will be present in every aquifer carrying layer at and below that level (assuming aquifers aren't disabled, of course). The soil thickness is at most 4 unless DF is hacked, and only a few types of rock carry aquifers (Conglomerate, for instance). This means most aquifers are 1-2 levels deep and found in soil only, but when you get rock aquifers they can be quite deep (more than 20 levels, and hitting the first cavern, but without leaking into the cavern).
Note that there are a number of sneakily named soils that sound like they would be Clay or Sand, but actually aren't, which means that the Clay sounding one actually doesn't block aquifers.