Deftone, I gave you a Like, but I must also give you a public acknowledgement for that OUTSTANDING identification effort.
I must also mention, the Ebay seller is wrong (as is so often the case)... it is not a Livery button. It is a Sporting (or Sporting Gentleman's) button. Livery buttons were/are worn on the uniform of household servants (butler, doorman, coachmen, etc) of Nobility/Aristocracy or merely wealthy families, and the buttons show the Family Crest or Coat-Of-Arms.
I should also mention, many Livery buttons do show an animal's head... but almost all Livery buttons also show a Torse, which is a wreath with six twists, viewed on edge and laying horizontally. You can see the Torse laying horizontally below animal's head on the (British) aristocracy Rowley family Livery button, below.
A Nobility family's Livery buttons tend to include a version of Coronet which denotes the noble's rank (duke, or baron, etc.) Note, a coronet is shaped differently than a Royal crown.