The DMG death domain is an optional rule.
What about the PHB Death Domain? Is it an optional rule too?
Not two mistakes, three in three different books all on the same topic. Highly unlikely. He also per RAW wasn't born a demigod, he became one, which defies the DMG classifications. Perhaps there are multiple classifications of demigod. One thing is clear, though. He's not anything other than a demigod, despite his presence as an optional rule in the death domain and being on Appendix B as a possible god(per RAW the DM decides who on that list is worshipped).
He's a demigod unless the DM opts to make him a full god. Any by the way, an Acolyte should consider domains when picking a god, even though he doesn't cast spells. The presence of domains in Appendix B is not proof that the god grants spells.
"If you're playing a cleric or a character with the Acolyte background, decide which god your deity serves or served, and consider the deity's suggested domains when selecting your character's domain." PHB page 293
Hey Max, when you bold things, maybe you should bold the RIGHT things.
"If you're playing a cleric or a character with the Acolyte background, decide which god your deity serves or served, and consider the deity's suggested domains when selecting your character's domain."
It is... really obvious that an Acolyte doesn't have a domain. So, you have two ideas here. Acolytes and the gods they worship and clerics and the gods they worship and the domains they choose.
Additionally, you are making a bit of a small error in your "three mistakes" claim. Your third is from Mordenkainen's and did you actually forget the conceit of Mordenkainen's?
"A companion to the Monster Manual and Volo’s Guide to Monsters, this book contains the musings of the renowned wizard Mordenkainen from the world of Greyhawk." pg 4
Mordenkainen is an unreliable narrator, and he could easily be WRONG about Iuz "becoming" a demigod. He could have easily been born that way. After all, it isn't like Mordenkainen met him before he was trained by his mother and was able to measure him and see that he wasn't a demigod.
So, he is referred to as a god in both the PHB and DMG. He grants spells and answers prayers like a god. And the only places he is referred to as a demigod are the writings of one of his enemies, and a summary of the setting which might have had a typo.