I’d agree with the first definition of privacy, but not the second.
Communications are only really private when messages and their metadata (the when, where, who etc.) of the communication are known only to the sender and receiver of the message, and not to any third party; like the postman, for example.
People using a communication service may perceive degrees of privacy, and perhaps consider that when they use a communication service that allows a trusted 3rd party to have a degree of information about a message and its metadata, they afforded privacy; but it falls short of what we should all consider private. Goodness knows how many people have just had this realisation thanks to FaceBook and Cambridge Analytica.
A considerable amount of information about an individual and their lives, and the lives of those they communicate with, can be gathered from metadata.
The classic example would be call records that show someone regularly dialling a number late at night which is not their spouse’s, then one shortly followed one day by a call out to an STI clinic, followed by the Samaritans. This information being accessed or processed by a 3rd party should rightly be considered an egregious violation of privacy.
Anonymity is related to privacy, but i’d consider it orthogonal.
I could, perhaps, have a truly private conversation with my wife on the dark side of the moon, but it wouldn’t be anonymous.
Likewise two individuals could have a publicly broadcast conversation, without their identities being revealed. Not private, but still anonymous.
It seems to me that the SAFE Network will provide—by default—private and pseudonymised communications (email or messaging for example), with both fully anonymous and public options available; should they be desired.
Jim