Hmmm...back when the 20D came out, it was routine for folks to have to dial in FEC +1EV routinely, in order to get the right tonal density in a scene shot with flash...regardless of direct flash or bounced flash it was the same. Then, when the 40D came out, that need for 'permanent FEC' disappeared...direct flash and bounced flash seemed accurate without FEC. Not sure what is the issue with your 5DIII, but offer that bit of historical insight.
The fact that you have already bumped up FEC with no better results might be a clue as to the nature of the problem. The shots you posted would not seem to have any corner case content that should fool ETTL metering; the white shirts in the first photo might result in a bit of underexposure, but certainly not the other shots.
The G9 shows you what you should get...my speculation is that the original Canon dSLR flash engineering team left Canon for a while, but have now come back to Canon.
This is great info and very interesting. Perhaps the engineers at Canon does change the algorithm of the way a camera body reads the E-TTL signals.
I took a few more shots this evening of and decided to use different ISO (as suggested by Tony earlier) and I started to notice something I've never noticed before. The 5Dmk3+580ex2 flash output seems to match the exposure set on the camera, no more, no less. It is as if taking the photo without any flash at all.
It also seems that the flash output commensurate with the camera's metering. If the camera's meter is under then the resultant picture with flash would also be under. At any one time, the background and subject looks as if it had the same exposure level. I.e, dark background, dark subject or bright background, bright subject. As I dialed the ISO up (more sensitive) the subject became brighter as did the background.
I will post some pic a little later to illustrate.