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 traces the history of the term 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)