"Coin Buttons" have been so named because of their proximity in size to coins. They are also called "flat buttons". But these brass buttons were popular because of their durability and beauty.
Most were gold plated and some were in silver plate. The process for plating metal had recently come out and became very popular with buttons.
Early in the 19th century, America wasn't manufacturing buttons enmasse. They were mainly imported from England and sold by wholesalers. Sometimes they were stamped with an eagle design to Americanize them for their customers.
They came plain or with beautiful designs. The back of the button sometimes bore the name of their gold gilt process such as "Treble Gilt", "Double Gilt", etc. and the name "colour" was a giveway to their British origins.
This button I recently found still retains much of its gold gilt.
The button is listed as "Andrews & Brothers" Philadelphia and dates to 1820. The company went out of business in 1828.
The company of Andrews & Brothers paid a button manufacturer to have their company name on the back of their buttons. This was a great form of advertising.
Lots of these civilian coin buttons found their way on coats and clothing during the civil war and are found frequently.