How do I stop Steam from sharing my library?
What is the problem? Only reason I can think of that you would see "borrow" instead of "play" in your library, is if you are Family Sharing the game from another account. So it would not be in your library, hence not seeing a "play" button but "borrow" instead.
How do I unborrow a game on Steam?
Do I lose my games if I uninstall Steam?
Why do all my Steam games say borrow?
Do I have to rebuy a game on Steam if I uninstall it?
How do I make my library private on Steam?