but they have to consciously activate their powers like healling
That would be more of a gameplay mechanic than anything, but the point is that he's shrugging off those bullets like their spitballs. It's not a matter of having a healing factor, it's a matter of how much pain and injury something like that inflicts on him, which is more about endurance than it is healing from the damage caused. We see Dante take and survive plenty of hits in cutscenes that are worse than some bullets from five-five-sixers.
And we see Dante and Vergil use their Triggers outside of Limbo, so we know DT isn't dependent on Limbo either.
Considering none of those points are really touched on in the game, Dante Shrugging off bullets but Lillith dying to one normal bullet (which shouldn't happen to a demon) I am going to chalk it up to a plothole.
We can hypothesize as much as we wish about "Oh, because of this maybe this is why it happened". But if in the actual game reasoning is given why regular bullets can kill a demon with ease, but bounce off a nephilim, I shall concede the point.
I gave you that reasoning - Lilith's human body is a vessel that she's tied to, making her as powerless as a human is in the human realm. Even Mundus' vessel "Kyle Ryder" was the same way, the only thing that kept him immortal was because he was connected to the Hell Gate, so his soul wasn't tied to the Kyle Ryder vessel. However, once Vergil closed the Hell Gate,, Mundus lost his connection to it, and it tied his soul to the vessel, which allowed him to be killed in the real world. He was still able to use what magic he had drawn from the Hell Gate prior to its closing (which was a ****-all great amount), which allowed him to turn himself into a steel and concrete colossus, but the human vessel of Kyle Ryder was still in there, and that's ultimately what got killed.
So...soul binding is the reason, I guess? Demons need to bind there soul to a human vessel (unless they have a Hell Gate to connect to) in order to operate in the human realm, but Angels (and most likely Nephilim because of that angelic heritage) don't have that limitation.
As I mentioned, that's most likely why the Nephilim are feared by the demon king; not only can they operate powerfully on their own, but they can also walk right into the demon king's realm and knock on his door.