It would be called a reel until about 5 years ago. As in silk reel, yarn reel (sometimes called a skeiner if used to make skeins, blocker). The Big Book of handspinning goes into this tools variations in some detail.
I use the larger version for yarn blocking here. (Chapters are in the description so you jump to blocking)
The skeiner or click reel should be in this video too. It's a newer version (1800s) of the reel you show in the video) oops. Different video.
Now the words are changing meaning. Anything that holds yarn is called a swift, apparently, no matter its function (swift is a very different tool, but it might help with google)
Old style, name follows function
New style, everything is a swift now.
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