The websites don’t nag you, actually. They just send you data. You have configured your user agent to nag yourself when the website sends you data.

And you’re right: there’s no difference. The web is just machines sending each other data. That’s why it’s so funny that people panic about “privacy violations” and server operators “spying on you”.

We’re just sending data around. Don’t send the data you don’t want to send. If you literally send the data to another machine it might save it. If you don’t, it can’t. The data the website operator sends you might change as a result but it’s just data. And a free interaction between machines.