The Presidential oath of office is a promise by the incoming president, to the people of the United States. The fact is, a private oath only is done to cross a "t" and dot an "i'. It has no legitimacy until made in public. And his public oath on Monday has more legal binding than his private oath if only because it has an overwhelming number of witnesses.
Isn't Obama the one who messed up his first oath of office 4 years ago?
And yes, the thought of "Koran" popped up too. Not that it matters. A false oath on the Koran is just as henious a sin in the eyes of the Lord as a false oath on the Bible. A false oath of office by the President is any failure to uphold the Constitution.