PHB p. 58: "You can cast a cleric spell as a ritual if that spell has the ritual tag and you have the spell prepared."
The same text for druid; you don't need to have it prepared if you're a bard or wizard, or using a book as a Warlock or with a Ritual Caster feat.
So (and I apologize for spelling it out), in the passage you cite (PHB 202), the "or" at the beginning of your bolded part sets out two possibilities, and the unless clause adds a third:
The caster must also have the spell prepared [in the case of the Cleric or Druid] or on his or her list of spells known [in the case of the Bard], unless the character's ritual feature specifies otherwise, as the wizard's does [and in a different way Book of Ancient Secrets and Ritual Caster does].
That's what I see at least.
The part that may get confusing is that clerics, paladins and druids do not have "Spells Known" they only have "Spells Prepared"
Then classes like Sorcerer, Bard, Warlock, and Ranger (per RAW I've homebrewed this) do not have "Spells Prepared" they only have "Spells Known"
I believe wizard is the only class that has both "Spells Known" and "Spells Prepared" so only Wizards and Tomelocks (or people who took the Ritual Caster Feat) can cast ritual spells without having to prepare them.
Is it a massive pain in the butt and frankly frustrating for Clerics? Yes, yes it is, but such is the RAW of the book.
We may have been doing it wrong all along ... not going to change mid-campaign but maybe next campaign.
