No, it's generally "Copyright <name>" to explain where the license is coming from, but a copyright license is the exact opposite of "all rights reserved".

No, not really. The phrase itself has no legal significance any more, anyway, but even if it did, this would basically work like an ACL:

    Allow MIT terms
    Deny *  # all rights reserved (implied)

Am I using it under MIT terms? Allowed. Otherwise not.

That is not in fact true on either point. Some legal significance remained for a while, and it is not nor was like an ACL.

* http://jdebp.info./FGA/law-copyright-all-rights-reserved.htm...