A ʿāmil is a grammatical governor that assigns a case relation; a maʿmūl is an expression governed by it.
In fī d-dāri Zaydun, fī governs the genitive al-dāri, while Zaydun is the delayed mubtadaʾ.
The advanced example coordinates across both a genitive and a nominative dependency, hence the need to track two distinct relations.