Do you seriously not comprehend the difference in their culture of the extent of a bow?
Someone above posted a brief description of the different levels. Try reading it.
Well, the source was about as clear as mud. On the one hand, a deep bow is used on formal occasions, such as a meeting of the President of the United States and the Emperor of Japan. On the other hand, a deep bow is indicative of subservience.
I could argue that Obama's deep bow was a formal bow but then I'd get a counter-argument that it was actually the bow of a subservient. I'd rather just cut to the chase and decide that anyone actually arguing the merits of the President's bow is a fucking crazy person and leave it at that.