"A self-signed certificate is one that is not signed by a CA at all – neither private nor public. In this case, the certificate is signed with its own private key, instead of requesting it from a public or a private CA."

This definition sounds right to me. Do you disagree with it?

I get what you're saying, that you can set up a certificate yourself. But you can't accept a certificate someone else set up. (in Chrome)

That's definitely the wrong definition. A self-signed certificate is a certificate that is signed by the private key corresponding to the public key contained in that certificate - in other words it's signed by itself.

In fact every CA root are by necessity self-signed and therefore signed by a CA (i.e. itself)

I don't think they were talking about certificates that were also CAs. It's not wrong, it's insufficiently precise.

Anyway, your definition is clearer, but it still supports the point I was trying to make. You don't "add your CA" in order to use self-signed certs. You do that to use a private CA that will sign all your certs. And doing so only allows you to use websites you signed, not websites other people signed. It would be a terrible idea to add anyone else's CA, and you can't easily use your CA to slap your signature onto websites other people signed. Adding your own CA is a completely different situation from trying to visit a self-signed site.