Vile Manifestation’s first ability counts only the cards with cycling abilities in your graveyard. It doesn’t care whether or not they were cycled to get there.

Vile Manifestation’s first ability applies only while it’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 Manifestation’s effect counts these cards.