You can cast domain spells in your arcane slots with just Mystic Theurge, you don't need the other stuff. If you have a level in arcanist, you can use it just as you say. Be aware you cannot use it with your cleric spells, even cleric spells prepared in Arcanist slots, as the arcanist ability applies only to spells prepared as an arcanist (even if they are in a different spell slot type).
Yes, the wording does mean that a cleric of the Light Domain gains 'Fireball' as part of the Cleric Spell List; and here's why the wording is different. I hope the following helps explain why
"The following spells are added to the warlock spell list for you," which shows that Fiend patron warlocks have Fireball in their spell lists, and so should be able to use Spell Scrolls of fireball.
The above Warlock phrasing, just as it's phrased for Wizards and other spell casting classes that gain extra spells through means of specialization, is phrased that way because each of those classes have a spell list that they must choose spells from as they level up. They do not inherently know all spells available to a Warlock. So the phrasing in this case means the fireball spell is added to the list of spells he can POTENTIALLY add to his 'Spells Known' list. For intents and purposes, those are now on his Warlock Spell List
Now enter Cleric, and the Paladin; which have very different methods of spell learning.
If you have a domain spell that doesn't appear on the cleric spell list, the spell is nonetheless a cleric spell for you."
It must phrase it like this because Clerics and Paladins do not have to choose what spells they learn as they level. Clerics and Paladins inherently have the knowledge of every spell on their list....because we're awesome like that....and can prepare spells as needed in between long rests. If I wanted to go with cure wounds one day and then give up cure wounds to take Inflict Wounds after a long rest? I can do that freely. I have direct access to every spell available to Clerics. So when I choose Light Domain and it says the Fireball spell that isn't on my class list nonetheless becomes a cleric spell? It's now part of my knowledge of cleric spells (On my list of spells), and thus able to be utilized in scroll form since I now have the knowledge of how 'Fireball' works magically.
The only reason it doesn't say these spells are added to your spell list for you, is because it wouldn't matter in a Cleric or Paladin's case. Any spell that's on our spell list is a spell we can cast should we choose to. Any spell that's on a Warlock/Wizard/Sorcerer spell list, is a spell they can cast only if they chose it from their spell list when gaining levels.
Best Answer
So divine source is powerful to you (as long as you have some neutrality). To followers, until you have tier 9, not so much.
You get to pick two domains. You have to choose any alignment domains matching your own alignment first, but if you have some neutrality, you get to pick at least one domain freely. If you are true neutral, you get to pick both freely. If you have no neutrality, you are forced to pick an alignment domain for both, and they miss out on the most powerful feature of divine source.
From these two domains,
You get to cast the spells from that domain whose level are not above your tier (so the 1st, 2nd, 3rd, and 4th-level spells from each domain at tier 4). Even if they aren’t otherwise on your list. That’s really nice; you get two new spells per tier that might have otherwise been completely inaccessible.
In addition, for each spell level that’s not higher than your tier, 1/day you can cast the spell from one or the other of the domains as a spell-like ability. That is a really big deal: spell-like abilities don’t have components, so feel free to grab domains that have expensive components and go to town.
(Note that the wording is unclear, and it may have been the intent that the spell-like casting is just how divine source lets you cast these spells, not a separate feature from the ability to cast them as spells normally. However, as-written, they seem to be separate and you get both the ability to cast these spells as spells from your normal slots, plus the ability to cast a spell of each level as a spell-like ability from its own daily pool.)
For followers, it’s a much less good deal. You might offer an unusual selection of domains (if you are true neutral), but that isn’t really all that valuable (you could be a cleric of an ideal for that). The cap on spell levels to your tier makes it a pretty terrible option for most would-be followers. Also, note that divine source gives followers no new ability to have domains in the first place: they have to have levels in cleric or something similar for it to mean anything.