Meredith,
The environment extent settings apply to the input and raster processing setup *before* the tools run. It's a little tricky to wrap your head around, but that's why Snap Raster does not work with Project Raster they way you are thinking. I think you think they are snapping but you've just been lucky and the outputs are just close.
The way to snap your output is to determine the lower left corner of the raster you want your output to line up with -- and use that location for the Project Raster tool's special Registration Point parameter. Then all will be well.