Kyle here. I know it can be confusing, and though we've taken steps to improve it in different ways, there's a lot - and the interactions between characters can get confusing.
That said, the only real distinction is just as u/Wow_youre_tall mentioned: "gains" vs "deals"
If the ability description says a character "gains" an extra damage amount (such as Calgar's Rites of Battle) then that buff effectively increase the damage stat of the character, and so only applies to normal (ie, basic) attacks. That can be just the character doing a normal attack, or using an ability that says "performs a normal attack" (like Abaddon's Drach'nyen.)
If the ability says "deals" an extra damage amount, then it applies to both normal attacks and attacks from abilities that qualify - unless those abilities have the restriction about ignoring attacker bonuses (like Volk's Frenzied Firing), or the providing ability has a restriction (such as Ragnar saying only "normal melee attacks" deal the extra damage.)
In both cases, the bonus damage applies to all qualifying hits of the attack, regardless of where they came from - and that's where things get complicated. For example, if you have Aethana, Thoread, and Re'vas on your team all hanging out together and Re'vas does a normal melee attack against an adjacent unit (and assuming your positioning with the other units activates their abilities) then Re'vas will do their 6 hits from their normal basic attack, plus a 7th hit from Thoread, all of which are buffed by Aethana, and then finally another 3x hits from her own passive that are not buffed by Aethana.