"It was my understanding that leger vs spiller had to do with unstructured vs structured play." True.
"What is the difference between leg and spil then?" Playing catch is "en leg". Football is "et spil". "Leg" is usually improvised and with looser rules and the people playing it are younger. The problem is that you have to use "game" for both activities since the english language does not distinguish between the two.
"spiller en leg" and "leger en spil" sounds bizarre to my native ears, sorry.