Patience would have helped!
There are supposedly ten entries for John Williams but only eight listed in the book. The reason I say this is because all entries with the same surname and christian name have a roman numeral next to the name, in this case (III) to (X). On the previous page the last entry is for David Williams, but no mention of JW (I) and (II).
The 1st entry for JW (III) shows him licenced in 1832 and as they are in date order, all the other JW's are at a later date.
As the Liverpool Pilot Service is unlike any other in the UK, they operate an apprenticeship system, your JW would have had to join the Service in 1806/7 and probably be licenced about 1811/12, which is feasible, but he could not have been a Master Mariner as well as a Liverpool Pilot.
I have looked in the 'History of the Liverpool Pilot Service' by John S Rees and there is an entry for a JW as one of the first licenced Pilots in the Port of Liverpool in 1766, but this is too early for your JW. Unfortunately there were no further lists of Pilots between then and 1832.
The only record available for further research is the 'Pilot Character Book' to be found at the Liverpool Maritime Museum, Albert Dock, Liverpool, where there is a page for each Pilot since 1766.
And there was you thinking it was just a simple 'look up'! Mind you with a name like Williams, nothing is simple.
Hope this is of some help