What is the proper name for a ship's crows nest?

The Crows cage

The crow was an essential part of the early sailors' navigation equipment. These land-lubbing fowl were carried on board to help the navigator determine where the closest land lay when the weather prevented sighting the shore visually. In cases of poor visibility, a crow was released and the navigator plotted a course that corresponded with the bird's because it invariably headed straight toward land, "as the crow flies."

The crow's cage was situated high in the main mast where the look-out stood his watch. Often, he shared this lofty perch with a crow or two since the crows' cages were kept there: hence the "crow's nest."

https://en.wikipedia.org/wiki/Crow's_nest

Often people confuse the crow's nest with the ship's top.

On a traditional square rigged ship, the top is the platform at the upper end of each (lower) mast. This is not the masthead "crow's nest" of the popular imagination - above the mainmast (for example) is the main-topmast, main-topgallant-mast and main-royal-mast, so that the top is actually about 1/4 to 1/3 of the way up the mast as a whole.

https://en.wikipedia.org/wiki/Top_(sailing_ship)