I did 2 screen prints captures. Please note the times. In the first one, he has his left hand down by his side while the king takes his right hand to shake it. In the second screen capture, he lifts his left hand to grab the kings hand
Not really quite sure what you’re arguing - do you seriously think Obama didn’t bow? - but the Queen of England was significantly shorter, and yet somehow he didn’t go through the motions of a full-fledged bow.