If I had to guess, up to 1911 Whitehead and Hoag made the inlays (celluloid over paper) and shipped to USPC who made the chip. Starting in 1911 USPC made both the inlay (celluloid on celluloid) and chip. The celluloid/paper inlay lifted over time. The other possibilty is that W&H made the chip.

I know that W&H made the flag, jockey and "L" (Lafayette College) inlays as I own the identical pinbacks. All of these chips are of the lifting inlay variety, just like the Anona and Pluto inlays.