"Angel Trigger"
This is how I see it.
When Vergil takes back his other half and seals off his heart, he, in his mind, becomes Pure. Godlike. A living Angel.
He's a corrupt Angel who has taken on the divine act of resurrection and the "divine" power of hell. He said that being a demon was "what power is all about" to Eva (paraphrased).
So, while Dante sees power as merely a means to achieve his goals (defeat evil and defend humanity), he becomes more "demonic" because he'll do anything to see it through.
Even if it means performing an Act of Pure Evil itself (slaying his own brethren, Vergil). He becomes a demon by sealing off his own Angelic traits to get the job done, no matter how monstrous he himself has to be in order to do it.
[An Act of Pure Malice itself, sealing off Dante's Angel side because he had to become the thing he hated most -- a Pure Demon -- in order to stop his brother from taking away humanity's freedom. A Devil of Enlightenment. A Light Demon.]
Vergil, on the other hand, initially wants to protect humanity, and is willing to do so at any cost, including his life; thus making him an official "Angelic Martyr."
[An Act of Pure Righteousness -- giving up your own life because you honestly believe you're protecting humanity. A Dark Angel.]
He dies, then journeys through hell as an act of Divine Retribution. He still sees himself, in the right, as an Angel, only this time, using demonic power to raise himself above all else.
Vergil then resurrects through sheer will power alone. A few humans have done so in the past (who have been declared dead), but Vergil is the only being capable of doing it consciously.
Thus, Vergil becomes a Pure Angel; aloof, seeing himself as completely in the right, and willing to seal off any emotion in order to achieve his own goals
[Meanwhile, Dante embraces the rage in his heart in order to achieve his own goals -- however, Dante achieves his Pure Demon Status through Angelic Power -- achieving Ultimate Goodness through an Act of Evil.]
[Dante hates the world around him, thus creating a "Quicksilver-World" DT -- Vergil hates himself because he felt unloved by his parents and believes that he needs more power in order to be "worthy" -- thus he creates a powered up, ideal mirror-image of himself to define his very being.]
In short, Vergil achieves Divinity through Darkness, while Dante achieves Depravity through Enlightenment.
Vergil is the Angel (God?) Trigger, while Dante is the Devil (Azazael?) Trigger.
[Yamato: Japanese Tradition and Obedience -- Rebellion: Rising Up against Overprotective Overlords.]
[Also, Vergil only calls it a "Devil Trigger" because he himself has misnamed it on purpose. In his mind, "Devil" sounds cooler, thus appealing to his Angelic vanity (as you can see in DmC, most demons don't care all that much about their looks). It's an Angel Trigger, pure and simple.]
Two Living Supernatural Paradoxes, ready to face each other down in DmC2.
*DmC2 gets cancelled*
NOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO. :'(