There are two sense of "privacy". Actual privacy, which of course is subsumed under anonymity, and the fake "privacy" that everyone is telling you they have a policy about and are assuring, while they track you and collect personal information.
Food for thought: Rather than nitpick on people who apparently use the word in earnest in its actual meaning and thereby conceding that those policies are actually "taking your privacy seriously", rendering the word void of meaning, we educate and take it back?
Saying "Make sure that it's impossible for my online actions to be traced to my identity, and then I don't need privacy" is in a way giving up on privacy alltogether.