I think the GP in this context means, e.g. LinkedIn wants their information to be public in the cases where it benefits them as a business. But then they want it to not be public when it doesn't benefit them. There is no such thing as "public information, except ..." - information is public, or it's not. If none of LinkedIn's data was public, they would have a much harder time getting people to sign up, and having as many users signed up as possible is part of their business model.

From a copyright perspective (since that's what LinkedIn's lawyers claimed): imagine if a newspaper sued another newspaper, saying that - not just the content of its paper - the information in the newspaper was copyrighted and could not be accessed by "unauthorized" third party companies. Either you print it, or you don't!