There is really only one true button quail and it is native to africa. Youve probably never see one. They arent kept in captivity in the US.
When you hear people say button quail or king quail they are referring to chinese blue breasted quail which are a very very small species of quail.
Anyone calling japanese quail button quail has no clue what they are talking about. Japanese quail are also know as coturnix, they are the only type of quail that were ever native to japan. They are much larger than buttons.
Japanese quail and button quail have the same general body shape but japanese quail are usually the size of a pigeon and chinese quail are about the size of a hamster.