The eagle in your early-1800s brass 1-piece flatbutton's backmark is "patriotic" -- actually neither Military nor Civilian.
Explanation:
In the very-early 1800s, nearly all of the brass buttons used in the US were imported from Britain, because the infant US button-making industry was not capable of producing more than a small percentage of the numbers needed by the Clothing Industry. But the war of 1812 happened, cutting off the supply during the war. When the war ended (in 1815), patriotic Americans boycotted British-made goods until their resentment cooled in the early-1820s. During that time, American button-makers put a patriotic eagle in their backmark, to distinguish their products from British-made ones. That being said, it is rumored that some unscrupulous British button-makers copied the eagle, to evade the boycott.
