Earlier quoted context omitted.
“Cookies” is shorthand for “persistent storage” because nobody outside of web developers knows other methods exist. When people, laws, banners, etc. refer to cookies, they mean “any technology that stores information on the client side systems”. Whatever mechanism is used is irrelevant to the meaning/concept.
I think of cookies as a mechanism to send data across the network. That mechanism can be used to simulate persistence on the client, among other things. But at least some of this conversation revolves around what the public perception of cookies is, and as for that, I really don't know. I wouldn't presume that anyone else knows either unless they've conducted a poll.
I can't get with that definition. A server that attempts to set a cookie is very explicitly asking for state persistence on the client in the otherwise stateless HTTP protocol exchange. It literally has no other purpose.