Sorry, I remember reading these in the email but then I forgot to respond when I got home.
I think these are quite interesting, I'm away from my PHB at the moment but I assume that the invocations affecting specific spells are based on the domain spells.
I'm not 100% sure about burning brilliance, it might be too powerful but then there are other spells which do the same so maybe not. At the very least I would consider a change in wording to allow the light priest to choose to make their light spells damaging so that they don't summon up a light source and burn their allies.
Trained in War fits the war domain perfectly. Are clerics going to have access to something like the warlock's 3rd level pact boons? If not, then perhaps another war invocation could be the one which allows them to gain Extra Attack.
The aura of life invocation might remove the need to spend hit dice during short rests since everyone can huddle around the life cleric. It definitely fits the domain though.
Yes, the spell related ones are tied to Domain spells, though I might also add one or two like the Arcana Divine I posted above, which give new spells as well.
I was mostly throwing around ideas. I thought of giving the Cleric something like the pact boons, as I mentioned in the "Extra layer of customization" bullet in my first post, but have decided that their free choice of the spell list does plenty for them, and I don't want them too similar to Warlock. That said, an invocation giving extra attack would not be out of the bounds at all.
I was a bit iffy about burning Brilliance anyway, I just wanted something that would reflect each domain, and immediately read like it was from that domain.
I am not too worried about removing the need for hit dice on short rests, people will still need to long rest eventually to regain spells. I was mostly worried about how it might play in combat, but figure that everyone clustering around the cleric for 3 hp per round will make a good target for AoE, so people will not be too encouraged to pile together. You might get a Pally/Cleric/Tank(Fighter or Barb) team that sticks together, but that is just tactics.
Most importantly, Interesting was what I was aiming for. Glad I hit that mark. Now I just need enough that these don't become the default picks for players, and they can still have characters that are interesting in a different way.