Here's how I do it. I have the AF-ON set to the AE-L/AF-L button, set on center-point, together with appropriate ISO and F/stop for my stationary bird pictures. This is saved into U1 user mode. In this mode the camera only focuses when I hit the AE-L/AF-L button. I only re-focus when the bird moves closer or farther away. Once focus is set it leaves me free to not have the shutter button attempting to re-focus every time I touch it or when I re-compose.
For fast-moving birds or animals, I set a somewhat higher ISO, preferred shutter speed or aperture and AF-C focus mode with dynamic autofocus. In this mode I leave the autofocus linked to the shutter button. This is saved into U2.
So I'll be using U1 for a stationary bird. If it should decide to take off I can switch the mode dial to U2 and keep shooting. Since the two user modes are directly adjacent to each other on the mode dial, all it takes is a small turn of one click to switch to the other user mode.
The reason I don't like AF-C on static subjects is that grasses, leaves or branches could be blown into and out of the picture and have the camera re-focus on that at the wrong moment. This especially happens on breezy days when the bird is among branches and foliage in a tree.