A reel is similar to a swift, but not the same. A swift winds a single ball of yarn. A reel is basically a standing rectangle with pegs on its corners to keep thread aligned; you use it to wind many warp threads for a weaving project, all of which should have both the same length and the same tension.

Videos of reels in use here:
https://youtu.be/Ba62vX5Zj1E
and
https://www.youtube.com/watch?v=PFBSKsOnwzI

I'm pretty sure the "rack" is a spool rack, so that you can easily use different colors of thread in a warp. If you watch
    https://www.youtube.com/watch?v=Uzetl5-sqS4
you will see (not very clearly photographed, I admit) a set of threads coming from behind the woman (on the left side of the image) onto her loom; the source of those threads is a spool rack.

A weaver might use a spool rack to feed thread to the reel, or directly to the loom. Reels are also called "mills".

-- Rachel Holmen, Berkeley, CA