I don't know if I really believe that.

As someone who lives around a few cats, I can confirm that. They wag their tail when they feel agitated, not just annoyed. Generally you also aren't supposed to approach a cat, but be in a position that the cat will approach you. That way you're less likely to cause it to dislike you.

Yeah, this is true. It's basically their way of telling you to back off, without going full on defensive mode with hissing and raised fur.