Their breakdown is what’s parroted up and down comment chains on this site when it comes to privacy/anonymity, so I’m frankly not sure how you’ve missed it over the years.
That, and the terms themselves tend to invoke clues about the meaning. Privacy implies there is an identity, but it is kept hidden. Anonymity implies there is no identity established so there is nothing to hide.
We don't see much of the latter since most web services require an email to sign up, at minimum, which still leaves discoverable bread crumbs. The web services that require you to give up nothing to use them are far less popular, so I guess I can see why people might conflate the two.
I'm not sure where you're drawing your implications from, but that is not implied, to me. I frequently see the concept of privacy applied to situations where an entity isn't required to ID themselves for the sake of privacy.
The common description when contrasting anonymity vs privacy is that anonymity allows one to do things publicly without being ID'd while privacy allows one to do things without the public having knowledge. There is no implication or requirement that the private party has been ID'd by another other entity.
