I usually use zone AF for BIF. Birds zip by too fast for me to use a single focus point, although I try that occasionally as well. Since zone AF can accidentally focus on the wings, I try to lead the bird a bit so that the zone is on the head or slightly in front. However, if I can't keep up (which is most of the time), at least it will focus on some part of the bird. For static shots, I occasionally switch to single point with expansion if I find that zone AF is picking up the wrong target.
The biggest eureka moment for me was realizing what tracking sensitivity really means. In the past, I always held down the focus button and tried to find the bird. In playing around in my house, I realized that holding down the button means that AI Servo will try to stick to the current target. So, if you start your focus on the wrong thing, like the trees in the background, then holding down the button means you are telling the camera to track the trees. With something like Case 1, where tracking sensitivity is set at 0, it will take a second for the camera to switch to something closer like the bird.
With this knowledge, it becomes clear how important it is to get AI Servo locked on the right target. If you realize that you are not focused on the bird, it's important to let go of the focus button and re-initiate focus. Doing this will force the camera to re-evaluate focus and it will switch to whatever is closest, which should be your bird.
To test this out, just use AI Servo at home and switch between any two targets. With tracking sensitivity set at 0, it will take almost a second for focus to change. With it set at 2, it will change much faster, but still not immediately.
Given that, it might sound unintuitive why I set tracking sensitivity to -2. First, I read it as a recommendation from Arthur Morris. More importantly, I tried a bunch of shots and realized that with high tracking sensitivity, if you miss the bird even once, the camera will quickly switch focus to something else like the background or water. With the tracking sensitivity set really low, once I got a lock on the target, focus would stick even if I temporarily lost focus on the bird. That means that I have many shots in focus even when the focus point wasn't on the bird at all momentarily.
Hope that helps.