What you have shown in your picture is not the clutch. It is the transmission shaft with the clutch pressure spring still attached to it.
The clutch assembly is the large round thing with gear teeth all around it that is attached to the end of the transmission shaft. The puller tool removes it from the end of the transmission shaft.
Now...... on to the transmission shaft:
The assembly is installed into the case with an interference fit. The holes in the case, called bearing bosses, are machined at the factory to be a few thousandths of an inch smaller than the diameter of the bearings. This gives the bearings a tight, non-slip fit into the cases.
To remove the shaft and bearings properly you need to heat the case either in an oven to approx. 180F/ 200F or if you have the experience, carefully heat the case around the bearing boss with a propane torch before pressing or driving the bearings and shaft out. Aluminum expands more than steel and the heating process causes the bearing bosses to loosen up their grip on the bearings a little bit.
NOTE: When working on the bearings while still attached to the shaft, never put any sort of pressure on the shaft or inner bearing race. Only apply pressure to the outer bearing race. Use a properly sized deep socket or short piece of pipe to accomplish this.
To re-install the bearings and shaft back into the cases do the heating process to the case again, only this time you also need to put the bearings and shaft assembly in a zip-lock bag and place it in your freezer it for an hour or two.
Freezing the shaft assembly causes it to contract or shrink, heating the cases causes the bearing bosses to expand. As quick as you can, re-install the shaft into the case The assembly should nearly push into the cases with your fingers. A little tapping might be necessary. It needs to be done quickly before the hot case warms the shaft bearings and the frozen bearings cool the case. If this happens the assembly will become much more difficult to work because the bearings will have warmed up and the bearing bosses have cooled. You will have lost all thermal advantage to properly re-installing the assembly.