can these words be synonyms?
(Not a teacher)
I would consider storeroom, storage room, and stowage to be synonyms (I've never seen a room marked "stowage;" it sounds like a nautical term).
A closet can certainly be used for storing items but many of these are personal items (clothes, shoes) that come and go so the function of most closets is more than simple storage. However, I think I have seen "storage closets." "Lumber rooms" and "box rooms," being specific, don't have the meaning of "general storeroom" but they are a type of storeroom.