Vile Manifestationâ€™s first ability counts only the cards with cycling abilities in your graveyard. It doesnâ€™t care whether or not they were cycled to get there.
Vile Manifestationâ€™s first ability applies only while itâ€™s on the battlefield. In all other zones, its power is 0.
Certain older cards have variants of cycling, such as basic landcycling or Wizardcycling. Vile Manifestationâ€™s effect counts these cards.