Several components can be involved
An open or damaged cap can admit rain at the flue. The crown, masonry, flashing, and chase cover can also be part of a water-entry problem. More than one component may be deteriorated.
Keep a simple record
Note when the water appears, where you see staining, and whether it happens only with wind-driven rain. Photographs from inside the room can help describe the pattern. You do not need to climb onto the roof.
Match the repair to the finding
A cap replacement addresses the flue opening; flashing work addresses the roof intersection. Mortar or crown repairs serve other purposes. A useful scope identifies the relevant components and the reason for the proposed work.
Avoid diagnosing from one stain
Water can travel before it becomes visible indoors. A proposal based only on the indoor stain may miss the entry point. Ask what was inspected, what could not be accessed, and whether further investigation is needed.
Further reading: Chimney Safety Institute of America. This general guide does not replace a property-specific assessment or the instructions for your appliance.