Charley, yes, there is surely some ambiguity in the wording! Along these lines, Hugill (in S from SS) gives two of his West Indian chanteys, both with "Heave Away, Boys, Heave Away." He says that in spite of saying "heave," they were both halyard chanteys. He goes on to speculate a reason for "heave" -- that they were originally used in cotton screwing!

Jerry, thanks for the cool reference. Quite amazing...

Q, thank you for forcing me to read closer. So in the passage you quoted, it does say "tug." And a bit earlier,
"Singing, or CHANTING as it is called, is an invariable accompaniment to working in cotton, and many of the screw-gangs have an endless collection of songs, rough and uncouth, both in words and melody, but answering well the purposes of making all PULL together, and enlivening the heavy toil. The foreman is the CHANTY-MAN, who sings the songs, the gang only joining in the chorus, which comes at the end of every line, AND AT THE END OF WHICH AGAIN COMES THE PULL at the screw handles. "
That's from The Merchant Vessel (1855).

So, tugging/pulling.
Moreover, it sounds like, from the description, that one would sing, "Fire, maringo, fire, a-WAY!" In other words, it is different from the typical halyard chantey way of coordinating the action, where the emphasis would be on "fire" (both times).

Another writer to mention cotton screwing, in New Orleans, in 1850-51, is Whidden, "Ocean Life in the Old Sailing-Ship Days":

"Each gang possessed a good 'chantie' singer, with a fine voice. The chorus would come in with a vim, and every pound in the muscles of the gang would be thrown INTO the handle-bars of the cotton-screws..." To me, this makes it sound like heaving, but again it is ambiguous.

Gibb