Option to Filter or Sort for Games You've Played
A new menu item in the my games and apps called "Played Games" where we can see all games ever played on the account. There should be a filter to see which games are disc based and backwards compatible (or in other words played on the account but not digitally purchased on the account).
-This way instead of remembering, guessing or keeping outside lists we can quickly see which games are backwards compatible on disc.
-Maybe we can also directly let the game be installed from played games and then after downloading put the disc in for verification to play.
BATMAN 7745 commented
You can technically do this under my games and apps, filter just games, then recently played and all 360 & Xbox1 games will be viewable in the order in which you played them.
Elijah Johnson commented
Using Groups, it's possible to organize your games into all those categories but I see what you mean wanting to make it automatic. I think integrating Groups into both Games and Apps would be an ideal fix. Add the automatic process if you want but I don't mind manually organizing my games into there.