Nowadays, a majority of apps require you to sign up with your email or even worse your phone number. If you have a phone number attached to your name, meaning you went to a cell service/phone provider, and you gave them your ID, then no matter what app you use, no matter how private it says it is, it is not private. There is NO exception to this. Your identity is instantly tied to that account.
Signal is not private. I recommend Simplex or another peer to peer onion messaging app. They don’t require email or phone number. So as long as you protect your IP you are anonymous
Anonymity is a very big part of privacy and always has been. That is why you don’t write your name on your voting ballot.
They are conceptually quite different.
People use both the terms interchangeably, but they are not the same thing.
Voting ballots are anonymous because you didn’t write you name on them (and they can’t be linked back to you hopefully), but they are not private because you have no control over how the data is used (once you submit a balot you have zero control over what happens to it next).
I’m not finding any definitions of “privacy” that suggest the term refers to control of something. Regardless of whether that something is within or outside of your reach.
From the page you linked:
Signal conceals what you say.
In a data sense specifically, I believe privacy refers to your data being hidden from unwanted eyes (aka you have control over who can see your data).
Which is also what you do when you vote. You control who has your identifying information and who has the information on how you voted. Which I guess is still different from Signal if we are still talking about that. Since you cannot control who has your identifying information.