If I play a Cull the Weak to kill my Immortal Phoenix and an enemy unit, can I play the cost to play it from the trash as the Cull the Weak killed an enemy unit?

Yes, you can play Immortal Phoenix from your trash in this situation.

According to Immortal Phoenix's card text: "When you kill a unit with a spell, you may pay 1 energy, fury rune (power) to play me from your trash."

Here is the breakdown of why this works:

1. Attribution of Kill: When you play Cull the Weak, it instructs each player to kill one of their units. Because Cull the Weak is the spell causing the kill actions, the kills are attributed to that spell (Rule 428.5.b).
2. Trigger Condition: Immortal Phoenix's ability triggers when you kill a unit with a spell. It does not specify that the unit must be an enemy unit; it simply requires that a unit was killed by a spell you control.
3. Multiple Triggers: If Cull the Weak kills both your unit (Immortal Phoenix) and an enemy unit, those are two distinct kill actions attributed to the spell. Immortal Phoenix's ability will trigger for each unit killed (as confirmed by the logic in Rule 428.5.d).
4. Resolution: Because it is a "you may" triggered ability, you choose to place the ability on the chain. Once placed on the chain, you pay the cost ([1] + Fury rune) as part of the resolution to play Immortal Phoenix from your trash to the board.

Sequence:

1. You play Cull the Weak. It enters the chain.
2. Cull the Weak resolves. You kill your Immortal Phoenix, and your opponent kills their unit.
3. Because you killed a unit (or units) with a spell, Immortal Phoenix's triggered ability goes on the chain.
4. You pay the required [1][A] (specifically [1][Fury]) to finalize the ability and play Immortal Phoenix onto the battlefield.

This interpretation is consistent with the cited RiftJudge FAQs (ID #6044 and #1631), which clarify that Immortal Phoenix can return from the trash when you kill your own unit with your own Cull the Weak.