Since Descourtilz's description pre-dates that of Beccari, Zombia antillarum (which is based on Descourtilz's description) has priority over Oothrinax anomala. In addition, Cook's name is invalid, since it was never formally described.
This means that the earlier account by Descourtilz has priority over Beccari's description, which in turn makes Zombia antillarum more authoritative than Oothrinax anomala. Moreover, Cook's name cannot be used because it was never officially described.