Maxperson
Morkus from Orkus
Absolutely. The DM can add or remove domains as he sees fit. I'd add Arcane to Azuth for example. Seems like Arcane would be a good fit for a god of wizards.The Appendix B domains associated with each god in the various D&D pantheons are explicitly only suggestions.
The PH and DMG overall read to me as saying individual gods have defined associated domains but that there are not official domains for any specific gods, just suggestions for the defined domains that a god has and therefore this is a campaign specific decision for each campaign, including one using a defined D&D pantheons from Appendix B.
So one DM using Greyhawk Vecna and saying he has the Trickery domain in their 5e Greyhawk setting campaign is as in compliance with official D&D rules as one saying he does not have the Trckery domain in their separate 5e Greyhawk campaign.
Similar to numerous aspects of 5e grid combat, 5e leaves it up to the DM.

The context though, is in examples of which gods have those domains. For instance your first page quote there gives examples of gods who are associated with that domain, while the second gives examples of gods who have influence of the domain. The non-exclusivity implied there is in the incompleteness of the list of gods in the examples, not whether the gods listed have those domains.PH page 58 "Each domain is detailed at the end of the class description, and each one provides examples of gods associated with it."
PH Page 59 "Each domain's description gives examples of deities who have influence over that domain."
"Examples" seems like it means a selected set of illustrative iconic examples and not an exclusive list.
Sure. It's a non-exhaustive list of gods, but as you note, it's up to the DM to add a domain to a god, not the player. The player only has the suggested domains to pick from unless the DM adds more.Page 60 under the life domain "Almost any non-evil deity can claim influence over this domain, particularly agricultural deities (such as Chauntea, Arawai, and Demeter), sun gods (such as Lathander, Pelor, and Re-Horakhty), gods of healing or endurance (such as IImater, Mishakal, Apollo, and Diancecht), and gods of home and community (such as Hestia, Hathor, and Boldrei)."
Page 62 "Gods of trickery-such as"
PH page 293 "Many deities in this section suggest the Life domain, particularly if they are closely associated with healing, protection, childbirth, nurturing, or fertility. As described in the chapter 3, though, the Life domain is incredibly broad, and a cleric of any non-evil deity can choose it.
A number of other deities, mostly evil ones, suggest the Death domain, which is detailed in the Dungeon Master's Guide. Most clerics who choose this domain are evil NPCs, but if you want to worship a god of death, consult your Dungeon Master."
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