All early Pierce bikes were just that, Pierce bikes. The headbadge and any graphics showed the name Pierce with an arrow through it, but not the word "arrow".
Even the first 1901-1903 Pierce automobiles were just Pierce. It wasn't until 1904 that the automobile division came out with the Pierce "Great Arrow", a large four cylinder touring car. After that point the cars were known as Pierce Arrows, and later in the aughts the company even changed it's name to "The Pierce Arrow Motor Car Company".
To my knowledge, the Buffalo produced Pierce bicycles (and the 1909-12 Pierce motorcycles) never had "arrow" connected to the Pierce name, although the symbol of the arrow through the name continued.
I, too, would be interested to see any Buffalo produced bike with original graphics that say Arrow, I don't think that exists, though.