Effect on current due to relative motion of load in homopolar generator
If the disk velocity in case 1 is zero, the current will be zero.
The total resistance in the loop that includes the load is the resistance of the load plus the resistances of the conductors connecting the load, the resistance of the connections of the conductors at the center and circumference and the resistance of the disk. Sliding connections at  the center and circumference will presumably be higher than fixed connection methods. The effect on current will be determined by the difference in the connection resistance compared to the total resistance in the loop.
For current to flow in the second case, the magnetic field must be prevented from passing through the moving conductors that connect the load between the circumference and the center of the disk.