Does Falling Star trigger Immortal Phoenix when it kills a unit with its reflexive triggered abilities?

Ruling: Yes, Falling Star triggers Immortal Phoenix. Although the reflexive triggered abilities are not themselves spells, all damage dealt by abilities is also dealt by the object that's the source of the ability, so the damage from Falling Star's reflexive triggers counts as damage dealt by the Falling Star spell itself.

Sequence:

• Falling Star is played and resolves, creating two reflexive triggered abilities on the chain
• Cleanup occurs after Falling Star resolves (no units die yet)
• One reflexive triggered ability resolves and deals damage
• Cleanup occurs; if a unit dies from this damage, it was killed by Falling Star (the spell) even though the immediate source was the triggered ability
• Immortal Phoenix triggers

Nuances:

• The reflexive triggered abilities themselves are not spells and resolve separately from Falling Star
• The cleanup where the unit dies is not "the cleanup immediately following that spell's resolution" (which already occurred), but the damage source is still traced back to the original spell
• This interaction works because damage inheritance is separate from whether the triggered ability itself counts as a spell