I don't know what a spinneret is in the context of numismatics, but the illustration that came up during your search is clearly a 19th Century draw bench. This machine was used to make final adjustments to the thickness of planchet strips and even their surfaces before the blanks were punched from them. It is of the type used at the U. S. Mint during mid-century, as it shows the suspended weights which returned the carriage to its starting place. This was an innovation devised by Franklin Peale, according to Roger Burdette's book From Mine to Mint.

I don't know what spinnerets have to do with the 1804 dollar or even coins in general, but I thought I recognized the term, so I did a quick patent search, which confirmed my initial guess. Spinnerets are used for forming filaments from viscid or viscous substances such as cellulose, by forcing of the material under pressure through one or more openings or capillary pores in the spinneret. They are used now to produce various threads and filaments out of plastic or carbon (for carbon fiber composites), but I suppose in the early nineteenth century they were used to turn cellulose into artificial silk.

A spinneret is the extrusion component used by the textile industry in manufacturing synthetic fibers like nylon, rayon, polyester and acrylics. The fibers are produced using the various processes discussed on the following web site: www.fibersource.com/f-tutor/techpag.htm

Nylon was the first fiber of this type that was a commercial success. It was introduced in 1939 as a substitute for silk and was used extensively by the military during World War II in such things as parachutes and clothing.

Current US "paper" money employs small quantities of synthetic fibers. The bills are printed on fabric consisting primarily of a 75%/25% mix of cotton and linen. However, they also have tiny blue and red fibers of various lengths evenly distributed through the fabric as a security feature. These colored fibers were originally made of silk, but were replaced by synthetic fibers some time after nylon was introduced. These colored fibers are probably produced using a spinneret extrusion process.