Yeah, Steve, I'm with you.

I "wish" the screwing cotton work style was the same as the shipboard work style because that would make a transfer from one context to the other very "neat" in explanation!

However, I personally am not bothered by this difference. In my opinion, the chanty FORM is the same. One just adapts their "performance" of the work depending on the task. That is even the case between the shipboard tasks where (at least as I argue) a "windlass chanty" and a "halyard chanty" are identical (or potentially so) while one simply works to the same song/form differently. Such is the case that *most* cotton screwing chanties and other stevedore chanties and rowing songs and etc (and a lesser number of menhaden chanties) have that same form.

If it only came down to cotton screwing work being different than shipboard work, that would be irksome, but when we consider the range of different kinds of work (and even non-work... or light work like corn shucking) across a consistent genre *form*, then we can make the argument that working one way or another is not embedded in a given song. Rather, the chanty form is a flexible one with respect to work method—hence my earlier rambling about not wanting to categorize songs (at least not firmly) in terms of a work task.
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One more suggestive piece of evidence re: cotton screwing working style:

Frank Bullen (1899 published, writing about the 1860s IIFC) gave this description -

"Below, operations commenced by laying a single tier of bales, side by side across the ship, on the levelled ballast, leaving sufficient space in the middle of the tier to adjust a jack-screw. Then, to a grunting chantey, the screw was extended to its full length, and another bale inserted."

"Grunting" is open to interpretation. Could it be that the song had some general "grunting" sound to it? Sure. But it implied more strongly to me that Bullen is referring to the "grunts" at the ends of choruses, as we hear in the stevedore recordings and as the 1830 writer (above) wrote as "whagh!"