That name has to come from some translation grammar issue. The first one was really a dissonant mess of "baditude". It opens with Trish impaling and electrocuting the main character and then hiring him for a job. I think they just were trying to be metal.
DMC3 was really the peak, I loved playing as Vergil. The game basically rewards you for using all the moves at your disposal to string together combos and explode out with crowd control moves. You can also play it by jumping around with the shotgun which is not satisfying at all. I thought DMC4 improved Dante by letting you use all the styles together, the menu hopping in DMC3 made it so I only ever used swordmaster.
Those without controllers need not apply. The higher difficulties make the AI extremely vicious and the dodging/spacing moves in this game are already very fiddly.