You have a "2 clause BSD" licence. What you call the resultant license is irrelevant. What matters is the text of the license. Heck, you could call it a "modified 3-clause BSD license, please see LICENSE" if that's what you want.
Come on, is saving face that important to you? "2-clause BSD" already has a well-accepted meaning. You might as well have said that GPL refers to "Gary's Perfect License"; After all, it's a GPL.