Is there a way to keep free games on the list just like paid games when I uninstall them? I want to keep my free games in the list so I know which ones I had when I reinstall my PC.
No, you cannot.
Free to Play games (which are not installed) will only appear in your library if you own any DLCs or have early access to a particular game. If you wish for a list of Free to Play games which you have previously played you will have to go through your games list through your profile.
This list will include any Free to Play games you have previously installed, even those which you have not played yet.