After the release of the elgif in Keres' bundle, I'm confident enough in this list that I can post it.

Note that the elgifs presented here are pre-release and may be changed by the time Season 2 finishes.

Ignis Aeterno Lv. 1:

"20% boost to all parameters of Fire types, negates all status ailments, 5% damage reduction from Fire, Earth types & greatly boosts Atk relative to remaining HP (0-50%)"

Ignis Aeterno Lv. 2:

"30% boost to all parameters of Fire types, negates all status ailments, 5% damage reduction from Fire, Earth types & hugely boosts Atk relative to remaining HP (20-80%)

The above elgifs are repeated for all 6 elements. The resistance is always to [own element] and [element weak to that element], so water gives water/fire resistance, light gives light/dark resistance, etc.

So the elgif, in general:

• 20% (Lv 1) or 30% (Lv 2) All Stats (to correct element)
• Ailments Null
• 5% Resistance to [Own Element] and [Element weak to that element]
• 0-50% (Lv 1) or 20-80% (Lv 2) ATK relative to remaining HP
  • The 20% minimum for Lv 2 means that the elgif gives 50% base ATK and up to 60% more based on HP remaining in practice)

Names:

In addition, though I don't really expect them to appear as season rewards, there were 4 other element-specific elgifs, some of which were presented with Summon Ticket bundles. These are not repeated identically across elements, and only exist for elements which have released or announced guild raid leaders:

Earth - Brave Assault

• 50% ATK to Earth
• 15% Crit
• Fill 5-10 BC when attacking (50% chance)

Thunder - Brave Ferocity

• 80% ATK/15% Crit to Thunder
• 100% Crit Damage
• 50% EWD to Thunder

Dark - Brave Shadow

• 80% ATK/15% Crit to Dark
• 100% Crit Damage
• 50% EWD to Dark
• Null EWD

Fire - Brave Combustion

• 80% ATK/15% Crit to Fire
• 60% Spark Damage
• 50% EWD to Fire

...the Earth one seems subpar compared to the rest. Gumi probably should've rebalanced it when they rebalanced Wannahon. As for the fire one... I guess we can expect to see it in a ticket bundle in a week or two?