The official defines the dark princess as 3. Look at the official website, /kagero3/
This series is divided into algebras like this.
PS 1: Shadow Prison: Seal of Engraving Hall (1 generation), Engraving Hall (the first work), Dark Lamp (rumor).
PS2: Shadow Prison: Dark Illusion (Second Generation)
PS3/PSV/PS4: Shadow Prison: The Dark Princess &; Shadow Prison: Another Princess (3rd generation, following the enhanced version, 3- 1 and 3-2)