Re: The Darkside
The post above addresses your question about how the dark side of Saturn's disk could be illuminated by reflection off the ring, in a process analogous to earthshine on the moon, as you pointed out.
The bright ring marking the planet's disk is somethihg else. Sunlight grazing the planet's atmosphere will get refracted around the "edge", giving rise to the edge effect as seen from Cassini's position in the shadow. If Saturn had no atmosphere, that ring would not be there.
In a total lunar eclipse (not an annular, where the sun's disk peeks around the moon's limb) there is no ring.