My bad! I deeply apologize for the oversight during the terminology check—yes, 'Oracle' and 'Prophet' are exactly the same thing. I will get that fixed.
Regarding the new Omens, the mechanic is very similar to the Vagabond's quests: there will always be 2 face-up Omen cards available. As soon as you fulfill one Omen, you immediately flip a new one to replace it. If the Omen deck runs out, simply reshuffle the discard pile to form a new deck.