So it sounds like the game that's been played won't appear in your games library, but the rest of your games will and will be accessible to at least one other person...
I should add I'm pretty tired, so maybe I'm misinterpreting what is being said.
any one of your family members can be playing from your shared library at a given time.
I really don't think that's the case. The wording on that phrase seems very specific to say that only one person can play from your library, not that one person can play any given game from your library at one time. This is still better than I expected, assuming that you can edit your family group easily. Most real world scenarios where people aren't looking to scam a couple extra free copies of games would be covered. It also leaves the potential for you and a friend playing the same game, even though I really don't expect that.
Still, it would have been great if Microsoft used this as an incentive for people to buy digital versions of games, require online authentication for those games, and retain the ability to play disc based games offline as much as you want. But that's a bit too consumer friendly for this console.