It should be mentioned that "sprite" nowadays does not necessarily mean the old school image rectangle. Advanced sprite techniques exist that allow for lighting effects, for example. There is also a technique that allows for animation by mesh morphing. The game made by using this is one about zoo animals; I currently do not remember its name. This technique is actually based on irregular meshes and hence would principally allow for an "overlapping" packing like in the 2nd atlas. However, that technique also requires a higher texel resolution than those shown in the OP to look good. (In other words, the 2nd atlas is probably not an example of this technique.)
Using irregular bounding boxes is also not exotic these days. Texture packers usually export a file besides the texture, wherein the clips are stored by name/index and texel co-ordinates. However, the usual texture packers still deal with AABB only, even if they support bin packing.
Please stop calling them “sprite sheets”. It’s an insult to the industry.
Well, this is interesting. In the world of perhaps not exactly professional 2D game makers there are many examples of tools and libraries that actually use the term "sprite sheet". There are for example Texture Packer, Zwoptex, cocos2d, and perhaps most noticeable Flash (at least in CS6). So the term is effectively introduced in the pre-industrial phase.