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To avoid any confusion 'The Lewes' does indeed refer to Capt. George's ship (although I find no record of it in Lloyds) and not to the Town of Lewes in Sussex, nor Lewisham (which may have been called ('The Lews'), both of which were accessible by barge but somewhat impracticable for a night out in London.
The reason they travelled by barge to the ship, rather than stepping off the quayside, was simple: firstly, there were very few jetties and quays in the busy Port of London; and secondly, a prudent captain would moor his ship in clear water - often some distance downstream of the Pool of London - and post sentries on the deck as a security measure. Not so much in fear of violence as a defence against the certainty of pilfering.