I don't think it has anything to do with demon powers, especially considering that if he has angel powers, those should show up too. Personally, I think its some kind of power- up rather than a second or evolved form. Considering the premise of the game, everything seems to evolve around a second or higher state of being, the world even has a concious of its own, so naturally the same premise would be used for Dante's new form. Here's somethings I would like to point out about this new form and why I wouldn't call it a DT:
First of all, Dante gains similarities to the Original Dante, gaining white hair and a red coat. However, this is different from other DTs not because of its appearance but because the form itself. Notice that the white hair and red coat are more like a covering or cloak of sorts rather than an actual physical change. Personally, I believe we're facing another "Nero- type" form here where something is providing Dante his power rather it being from his genetics. Even moreso, Dante's DT and "Angel Trigger" are displayed in his weapons, so there's really no need for an actual DT.
Second, the way that Dante's new form changes combat is very different, as we have seen. But I would like to state two hypothesi on how these changes affect Dante and the World: 1) It could be merely a mental thing, sort of like a super adrenaline rush, where Dante taps into a certain state of mind. Or 2) Dante could really being the world around him. Assuming that this is some kind of metaphor for contradictions in the human state of mind, then maybe Dante's new form could have him mentally affecting everything around him.
Lastly are the powers themselves. Dante's new form is much less unique than the DT from the original series. Instead of him speeding up, everything else slows down, which I've seen common in other games. What's unique, however, is how the new form is lengthened which seemingly is by doing a crap load of aerial combat.
So, really, it would be more appropriate to call Dante's new form just "Trigger" rather than "Devil Trigger," especially since the reason by the name "Devil Trigger" is to imply that Dante, being half demon, is a conduit for the demonic powers of the weapons he carries (He is the "trigger" for the "devil") which is why in DMC1 we don't actually see his real DT (in fact, you can say that his real DT is actually him triggering Rebellion's power, especially since he didn't carry it at all in DMC1). However, things aren't set in stone yet, so maybe there might be some changes.