I switched to using the AF-ON button about a year ago as it allows me to use all three AF functions (AF-S, AF-C, Manual) through this method without having to change other settings.
For AF-C just keep your finger on the AF-ON button and you will be able to track moving subjects.
For the AF-S function use the AF-ON button to focus on your subject and then release, you will be able to re-compose if you wish.
Manual is exactly as it says, don't press the AF-ON button and focus manually. In all of the above half pressing the shutter release will not make any difference. One thing to note, if you are using a Nikon camera fitted with a VR equipped lens the VR will not kick in until you half press the shutter.
Using AF in this way took a little bit of getting used to but I wouldn't go back to the old method.