There's a pretty big difference between "private for the businesses, but Facebook continues to know All, as is usual for them" which is what you basically describe, and "actually private, including from Facebook" which is probably what a reasonable person would understand from "designed for privacy."
Well I mean Facebook kinda has to know based on how this works. It seems like the beacon essentially instructs the app to grab data for a specific page and display it. Based on this Facebook will know that a page was requested. Page data changes, it wouldn't make much sense for the button to store that data. The unique id however for a page wouldn't be changing, so that is what the button uses.
If you're that concerned about such a feature it is easy enough to not use the app, or simply not use this feature of the app.
You're probably right, but it doesn't suddenly become OK to lie just because the truth is evident if you think about it. Especially when the average person does not have the knowledge needed to reach that conclusion.
I think you're thinking about it more from the technical persons view of privacy. Most non tech people do not think about these kinds of privacy concerns or worry about what Facebook knows about them. They already have likely shared a ton of data with Facebook and don't care and likely don't care to share more. However they may be concerned if they found out every business they go to was getting a bunch of personal information about them.
So I think the general consumer would be more concerned about all these random places they go knowing who they are and details about them. This is what I believe Facebook is trying to state, that no business gets any of your information. Not that Facebook doesn't get anything. So yes they poorly worded what exactly is private and what isn't.
Do you think the average non-technical person interprets "designed for privacy" as "Facebook gets information about all your movements"? I wouldn't think so personally.