AFAIK, steam adds games to installed programs list when installed, which allows me to add them to my Xbox Games list (and record etc. without stutter etc.), but my Windows 10 install wiped my system (I had backup of games) and I got it back, but as it isn't installed on this installation of Windows, it doesn't see it, how can I get them to display on programs list, allowing me to mark it as a game.
Steam (and possibly other similar clients)
You should be able to move your games to the proper location ("SteamLibrary\steamapps\common\ [Game Name]" usually) and installing them again from Steam.
Normally, in this case, Steam will recognize that the files are already present and won't download them again. Instead it will run some small verification routine.
If you have games that weren't installed through Steam, try running once and see if they show up, but most likely you'll need to install them again.