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Everybody Hates Hades: Chernabog was a black god, but wasn't evil as a pre-Christian Slavic deity. Subverted as Walt said he was meant to be Satan anyway, and in the roadshow version, Deems Taylor identifies him as such.
Slasher Smile: When he really liked his final transformation of his transformed minions.
Villain by Default: Did Chernabog actually do anything inherently wrong? The answer - no, he did not, because he left no lasting damage and disappeared without having harmed anyone - apart from some demons perhaps. He summons an entire legion of demons, ghosts and otherworldly monsters so he can basically have a rave.
It is implied by Walt and Deems that he has done many crimes against God, so it is possible that he has harmed millions of lives before being sealed in the mountain. It's also possible he's enslaving and torturing the demons and ghosts he summoned.
Big Good: Against the Firebird's Big Bad, it seems a very dominant force of nature in a positive way, whereas the Firebird enforces nature's power in the negative way. In the alternate ending, it even becomes the sun (or perhaps sunlight).
Nice Job Breaking It, Hero: Well done, accidentally unleashing the Firebird upon the forest! (However, as noted at the film's fridge brilliance page, the forest is able to grow back better than before, so breaking things in this case may have been needed to allow previously dead ground to flourish again. This would fit in well with the death-and-rebirth theme of the segment.)
No Biological Sex: But since it looks female (or really, really Bishōnen), we're just putting it with the other Disney heroines.
The Lancer: For the Spring Sprite, helping her back onto her feet and assisting the renewal of life after the eruption.
Adaptational Villainy: The Firebird of the original ballet the song originated from was actually a helpful creature that aided a prince to defeat an evil wizard. This Firebird on the other hand is a creature of pure destruction.
Avian Abomination: It's really just a mass of lava that is usually shaped like a bird.
Blob Monster: The Firebird is an amorphous mass of lava, a trait which it uses to its advantage. For example, it tries to devour the nature spirit first from ground level and then ambushes her from the top of a cliff a couple of seconds later, manifesting its head both times.
Eldritch Abomination: Where as Chernabog has at least some identifiable motive and has a generally defined form, the Firebird is little more than a chaotic force of nature appearing as a jumble of flame and lava in the loose form of a bird.
Evil Counterpart: To the Nature Sprite; she embodies life and nurturing of the world, while the Firebird brings only destruction and death. The commentary prior to his segment acknowledges this by calling it "a tale of life, death, and renewal," with the sprite representing life and the Firebird representing death.