Originally Posted by JCHandy
...it appears that they have done, is cut a 45 degree end and then flipped the cut that came off and glued upside down to the end...
You're very close, rather than flipping it upside down, it's a normal outside miter. The piece that's on the end of the run is the same as a long run, but cut short, the thickness of the quarter round.
Rather than trying to nail that piece in place, get some tacky woodworkers trim glue
, it works really well, and doesn't split the small parts.