First method:

The finger will experience an upward force called buoyancy; the gravitational burdon of keeping it up is eased. There must be an equal and opposite force pushing up, including through the scale. The scale goes down due to its increased load.

Second method:

The water level rises in the cub. The pressure at and on the bottom increases with the depth of the water, no matter the ins and outs of the shape of the container or objects within the water (such as the submerged portion of an ice cube). The area of the base remains the same so the increased pressure is applying more force. Again the side with the affected water goes down.