It can be both
There is no reason these explanations must be mutually exclusive. Since there are places in the PHB supporting both of them, both can be true. Some powers are granted, and some are harnessed by the hero him/herself.
From the Warlock description, PHB p. 107 (emphasis mine):
... your otherworldly patron bestows a gift upon you for your loyal service. You gain one of the following features...
the same page:
In your study of occult lore, you have unearthed eldritch
invocations, fragments of forbidden knowledge that
imbue you with an abiding magical ability
From the Cleric description, PHB p. 59:
As a cleric, you choose one aspect of your deity’s portfolio to emphasize,
and you are granted powers related to that domain.
p. 58:
you gain the ability to channel divine energy directly from your deity, using that energy to fuel magical effects
In FR the dragons were created by the primordials. Does this mean many/most/all dragon gods are also primordials, as aspects/fragments of Io?
The Dragon Gods are presented as actual gods rather than primordials (DMG p. 10, PHB p. 296, & SCAG p. 113).
Can a primordial also be (or somehow become) a god?
I'm unaware of this occurring, but I wouldn't rule it out:
- Asgorath/Io is both, but (as creator of the universe) doesn't count.
- Tharizdun is close, but is not a Primordial.
- Kossuth is also close, but he's "not a true god but actually an
elemental primordial".
Do primordials need, desire or benefit from followers?
Not normally, but Kossuth is an example of a primordial that benefits from extensive worship.
Do surviving primordials have significant religions associated with them, either in Abeir or in Toril?
On Toril the five Elemental Lords have followings, while The Seven Lost Gods were once worshiped. I believe Abeir would have more significant Primodial worship, but that setting was never detailed.
Do primordials grant spells in the same way as gods?
Not normally, but again: Kossuth is an example of a primordial that both grants spells and has extensive followings.
Can they be warlock patrons?
I don't see anything preventing this. A few homebrewed examples of this can be found here and here.
Is there any obvious difference in the portfolios of primordials versus gods? I would have assumed that since they predate mortals they tend to represent more fundamental forces (like the elements) rather than human concepts like law, love, luck, agriculture.
That might be one way to view it, but 4e's creation myth (Worlds and Monsters p.56, cited below) has the Gods being created at the same time as the Primordials. Another way to view it might be that the Primordials are composed of elemental "physical-matter" while the Gods (and Astral Sea) are more composed of thought (mental-matter).
In short, what is the difference between a god and a primordial? Are they fundamentally the same (just different lineages) or are there fundamental differences?
They are consistently presented as having fundamental differences. From 4e's Worlds and Monsters (p.56): "The gods, beings of divine power, appeared in the Astral Sea, while in the Elemental Chaos arose the primordials, incarnations of tremendous elemental might"... "composed partially of creation-stuff".
I also found this fantastically detailed post on Candlekeep...
That still seems like as good a guide as any.
Best Answer
Yes, it's possible, whether or not the fiend and the god/goddess get along. There is no RAW prohibition from the multiclass, no matter how awkward it looks.
This depends on how the DM plays out the deity's and the fiend's role. Points to raise as you two flesh this out (a collaborative effort between you and the DM):
Playing an internally conflicted character is rich with role playing possibility. Trying to serve the good, still dealing with evil ... that's a standard story IRL, and an element of stories from many different cultures. This would allow the link with the Pit Fiend to remain -- but there's always a price to be paid! (This could provide DM some fun, occasionally at your expense! :) )
Why not? Here's a point of view that the Pit Fiend could take:
"I've got my hooks into this gnome, I won't let him go. So he wants to get in with that goddess/god? Good! I can use him to cause trouble for (chosen deity), and then enjoy his suffering as he realizes that it's his fault -- due to his hunger for the power only I can give him. Muahahahahaahaa!"
or
"I don't like (deity), but we both have (unrelated) unfinished business with that #@!^%$ Orcus, and this gnome will serve nicely as a proxy. In the end, I profit! Muahahahaahaa!"
You and the DM should be able to work out a deal, and some tension, that fits the campaign.
One last point to address as you work with your DM on this multiclass:
How important is alignment in this campaign?
Alignment in 5e is a bit different than in previous editions. It is more like an ideal for the character, and a matter of how the player should behave. How well the player character lives up to it, and what the rewards/penalties are, lay in the realm of DM's discretion.
With that in mind, and the two deities to choose from: Chauntea's clerics appear to require a serious commitment, and a decision on being a Pastoral or a True Shaper (the latter looks like a better fit for an adventurer), while Baervan's yoke looks like it is lighter to bear for an adventuring cleric.
Could the pit fiend and the god/goddess get along?
That would require a serious threat that both of them want to counter, something opposed to them both for different reasons ... so that you would be serving both of their interests (roughly) at the same time. Some existential threat to the world itself ... that too lies in the realm of how your DM is running and shaping the campaign.