The 560III have receivers for the 602/603 built into them. They are only dumb receivers - will only trigger the flash, not control any other feature.
The benefit of this is that you don't need extra triggers underneath them - no loose connections, batteries etc etc. Two or three years ago, this system would have been dogs b*****x.
An on camera 622 will not trigger the in-built receiver in a 560III. You will need a 622 attached underneath for them to fire from the on camera 622.
You could stick a 602/603 on top of the camera mounted 622, and then they will fire. I have done this with my 622 and 602 to fire remote guns, because I have a lot more 602s than 622s. However, if starting out now, I wouldn't bother with anything but a 622 and purchase flashes that are compatible with them - namely mkII Canon guns or 568 YNs et al.