Egg and Royal are two islands. Columbus does not specifically say that there are two islands in the log. Eleuthera Island is marginally visible from the Egg Island anchorage. There is no surrounding reef at Island I. Island II has no coast running North-South. Even if there were such a coast, it would not face Island I. Island II must be at least 5 leagues north to south.
New Providence is much smaller.
Island II must be at least 10 leagues east to west. New Providence is much smaller. That course from Conch Harbor puts Columbus aground on Andros. Columbus sailed ESE on the night of October 17. That course from Andros puts Columbus aground on the Exuma Cays.