The following extract from the Oxford Dictionaries blog says that it is from the late 18th century, and precisely from the a ship diary where surfing was recorded in reverendo to
The vocabulary of surfing in the English language has a long history. In 1798 Ebenezer Townsend, Jr., the supercargo (a representative of the ship’s owner) on the sealing ship Neptune, sailed into the Hawaiian Islands and recorded in his diary of 30 August, “They [sc. the natives] sometimes make use of surf-boards. The surf-board is about their own length and floats them lighter.”
This is the OED’s first documented evidence for the word surfboard. However, the OED cites an earlier account of surfboards in Hawaii, in 1784 by Lieutenant James King, commander of the HMS Discovery – but he uses the word plank (still used today to describe a type of heavy surfboard): “If..they should not be able to keep their plank in a proper direction on the top of the swell, they are left exposed to the fury of the next, and, to avoid it, are obliged again to dive and regain the place, from which they set out.”
The Phrase Finder cites a similar origin, precisely a ship diary dated 1779 from James Cook expedition to Hawaii:
It is generally accepted that surfing was first enjoyed by Polynesians in Tahiti and Hawaii. The oral tradition of passing chants, called 'Meles', from generation to generation records the practice of surfing in those islands and suggests that it dates back to at least the 15th century. It was first recorded in print in the ship's log of the Discovery, which was part of Captain James Cook's ill-fated expedition to Hawaii. The log was written in 1779 by Lieutenant James King, who took over the captaincy of the ship after Cook was killed.
But as you show in your question, the are a number of sources that suggest a much earlier origin of the term. The following extract from Surfer Today suggests that the term may ultimately derive from surge and tries to trace its history back to Latin surgo/surgere:
Interestingly, linguists believe that the word "surf" has its origins in the late 17th century, apparently from obsolete "suff", meaning "the shoreward surge of the sea". The language specialists underline that "suff" might have been influenced by the spelling of "surge".
So, now we've got "surge". This word dates back to the 15th century and can be translated as "a sudden powerful forward or upward movement, especially by a crowd or by a natural force such as the tide."
We can see (and hear) that there's still a logic connection with the sport of surfing. But the history challenge is not yet won. Let's dig a bit more. "Surge" (meaning fountain or steam) comes from Old French verb "sourge" which, in turn, is influenced by the Latin "surgo/surgere" (to rise).
*Linguists highlight that the word "surge" was initially used to reveal the "rise and fall on the waves&," and to express a "swell with great force," as well. The original Latin "surgo" tells us "to rise, arise, get up, stand up."
In the end, it all makes sense. Surfing involves humans "rising and standing" on a surfboard, but waves and tides also rise. We're stunned by what we found: the word "surgo," the linguistic mother of "surfing," has roughly 2,000 years.
Surgo/Surgere (Latin) > Sourge (Old French) > Surge/Suff (English) > Surf (17th Century)