If I understand the original post clearly, Gshocked, you are asking which AF Case some members here are using when shooting birds in flight with their 5Dmk3.
I can only speak for myself, but I prefer to control the 3 parameters that constitute the AF Cases myself, rather than relying on one of the presets which is what the 6 AF Cases represent.
That way, I can better understand the changes that are brought about by adjusting an individual parameter. I try to change only one at a time for the same reason. The three parameters are Tracking Sensitivity, Accelerate/Decelerate Tracking, and AF Point Auto Switching.
The settings to weight Ai Servo autofocus either toward Release or Focus Priority are also pretty important.
Then there are the different AF Point Area modes and how you choose to employ them.
I have written quite a couple of fairly extensive posts on my website detailing possible AF set-up options for the 5Dmk3, as well as the 1DX, as there is significant overlap in how the AF systems work. Links are here:
'Birds in flight' is not necessarily a one AF situation description though, as you get birds that fly fast, some that fly slow. Birds that are flying in a bright, uniform background like the sky are typically much easier to focus on than a bird flying lower down, against a dark or structured background (where the best exposure might be found). I find I get better AF and sharpness results when I have direct light from a low angle, illuminating the bird from the side and lighting up the underwing, but that light only happens very early and then again late in the day. It is also important to take into account whether the bird is flying "across" your frame, at right angles to the sensor, which I find easier to focus on, or flying "toward" you, more difficult.
That said, I typically shoot a 5Dmk3 with an EF 500f4L IS ii for birds in flight, and my basic setup is Ai Servo, Ist Image Release Priority, 2nd image Priority the middle setting (50/50), Ai Servo Tracking Sensitivity set on -1, Accelerate/Decelerate on 0 or maybe +1 depending on factors explained above, and AF point Auto Switching on zero. I typically shoot with Single AF point if the bird is similar to the background, or flying against a structured background, and switch to AF Zone if the bird is in a clear sky.
I also use the 300f2.8, and 600f4 on occasion.
The images below this post are all shot with the 5Dmk3.
The carmine bee-eater pair, EF 300f2.8L IS, Shutter speed 1/3200 at f8, iso 800
The crested caracara, EF 500f4LIS ii, Shutter speed 1/4000 at f8, iso 640
The pied kingfisher flying with fish catch, EF 600f4L IS ii, Shutter speed 1/5000sec at f5,0, Iso 500
Lovely images and thanks for the detailed information