Well, Ring of Regeneration is based on the Spell Regeneration, and not on Regeneration (Ex) which has two different effects.
The spell heals damage magically and therefor stops bleeding.
Anyway, i think i'll rule in my group that Regeneretion (Ex) stops all Bleed, even though its not magical healing. Less Book-keeping, and the different examples above suggest Regeneration and Fast Healing effects does stop bleeding.