You actually dont really need the tool for either but does make getting empties out of the clips easier. When loading the half moon clips I find it easier to load the two end rounds then put in the center one. That may not be the same for other folks but it seems easier to generate force going into the center slot. It also help to slightly turn the rounds as you are pushing them in, sorta rolling them into the notch.
The moon clips seem easier to get shells into than the half moons. Some of the other old time users can chime in. I have not messed with moon or half moon clips since high school when one of my teachers would take us shooting using a webley converted to 45 ACP using half and full moon clips or 45 auto rim.