Eponym link: nostril closure equals Young's operation equals atrophic rhinitis.
In atrophic rhinitis the nasal mucosa, glands and turbinates waste away, leaving wide cavities that collect foul smelling crusts. The treatment idea is to make the cavity smaller so drying air no longer reaches and dries the mucosa, allowing it to recover. This is achieved by surgically closing the nostril: complete closure is Young's operation and partial closure is the Modified Young's operation.
The other rhinitis types listed (allergic, vasomotor, occupational) are managed with medical therapy and allergen control, never with closure surgery. So the condition where partial and full closure are performed is atrophic rhinitis.