tl;dr “Privacy” = the data is private i.e. only on your devices. Or if the raw data is public but encrypted and the key is private, I think that qualifies.

“Anonymity” = the data is public but not linked to its owner’s identity.

If you’re sharing your data with a website (e.g. storing it unencrypted), but they promise not to leak it, the data is only “private” between you and them…which doesn’t mean much, because they may not (and sometimes cannot) keep that promise. But if the website doesn’t attribute the data except to a randomly-generated identifier (or e.g. RSA public key), the data is anonymous. That’s the article.

Although a server does provide real privacy if it stores user data encrypted and doesn’t store the key, and you can verify this if you have the client’s unobfuscated source.

Also note that anonymity is less secure than privacy because the information provides clues to the owner. e.g. if it’s a detailed report on a niche topic with a specific bias and one person is known to be super interested in that topic with that bias, or if it contains parts of the owner’s PII. But it’s much better than nothing.