So, I want to make a beta version of the game playable only from some beta testers that are not the creator of the game. But I don’t really understand how does the Restricted visibility works.
Restricted visibility is what you want. Set your page to that, then head to the “Download keys” section on your game’s edit page and generate keys for the people you would like to give access to. Then send them the links that are generated and they will have access to your game.