Lightning Phoenix's last ability looks at how much damage was dealt to opponents in the turn, even if it wasn't in your graveyard when the damage was dealt. It doesn't care if opponents gained life, even if they gained more than they were dealt damage.

Lightning Phoenix's last ability triggers only if it's in your graveyard as your end step begins and only if an opponent was dealt 3 or more damage before your end step begins.

If an opponent is dealt 3 or more damage and loses the game, Lightning Phoenix's ability triggers as your end step begins.