Comparing your camera meter to a lightmeter isn't necessarily going to give you the same reading (or even close). As Mike says, an incident reading will often be very different to a reflected reading. Even assuming you used your lightmeter in reflective mode, there are still big differences possible due to angle of view. If your meter is set to a fairly wide angle, and your camera is reading from a fairly small portion of the scene, they are seeing two very different things. In addition, the camera then adds its own algorithms to give its idea of the "correct" exposure, a lightmeter won't do that, it leaves that to the photographer.
It is quite likely that the 5DIII algorithms do aim to underexpose to a degree, many DSLRs do, in order to minimise blown highlights. Again, a lightmeterwon't do that for you. Different cameras use different algorithms, so two different models of camera won't necessarily give the same suggested exposure, and they will both probably differ from a lightmeter most of the time.
I am often shooting with a little +ve EC on my 5DIII (or the meter to the right of centre, if in manual) but yesterday, I was in bright sunlight shooting racing cars and I was in -ve EC all day long (up to a stop and a third at times) to keep the whites from blowing out. So, this camera (which is "underexposing" byt your definition) was shooting over a stop MORE underexposed to get the correct exposure at times. I was generally shooting the white / light cars for a couple of laps, then shifting the exposure to shoot the darker cars. If I shot the darker cars at over a stop -ve EC, then yes, they were underexposed, I shot those at -1/3 to 0 EC get a good exposure, but at that exposure, the white cars blew out badly.
As has been said above, the meter is a guide to help you assess the correct exposure. The camera doesn't underexpose, the photographer does. Sometimes you need higher than the suggested meter reading, sometimes lower. The camera makes a suggestion, that is all, and probably errs on the side of underexposure deliberately to avoid blowing highlights. It is for you, the photographer, to interpret that suggestion and obtain the correct exposure which may be a couple of stops away from the meter reading, in either direction.
I don't understand the difference between Av and manual of over two stops. If as you say the shutter speed, aperture and ISO were all the same in both shots, then the exposure would be the same, so I don't see how that could happen. Metering wouldn't affect that, EC wouldn't affect it either (as you say you took the shot in Av (so the EC would have been applied) then used the same settings to shoot in manual.
I don't even understand what you were attempting to show wioth that test either. If your metering is "wrong" then it would affect the Av shot, yes, but then using the identical settings in manual ignores the meter completely, so should be identical.