My cumulative summary up to this point.

1780s-90s:

General references to African and New World Black work-songs, from Mali, Grenada.

1800s:

General references to African-American work-songs and their style, from Martinique;
Rowing songs from Georgia, South Carolina, Guyana, Surinam;
Windlass songs, aboard vessels with sailors incl. from Northumberland and Holland.

1810s:

2 stevedore songs from Jamaica that resemble chanteys;
African-American rowing songs from Antigua, Virgin Islands;
Singing and fife-playing at the capstan on a British war ship.

1820s:

Rowing songs, from Georgia, Virginia, St. Thomas;
A version of "Cheerly Men" for topsail halyards on a brig near Quebec;
Fictional capstan shantying in the Arctic; capstan (?) song of British tars in London; chant for pulling known to an ex-British navy man. [I've also seen another reference to the phrase "British capstan song" from 1825.]

1830s:

African-American rowing songs in Florida, Georgia, North Carolina, South Carolina, Guyana, and "West Indies"
African-American firemen's songs on the Ohio and in general
Cotton stowing songs in Mobile Bay
"Ditties" at the capstan on an East India Company ship
A song at the pump windlass on a transatlantic voyage
Songs "for capstan and falls" and for catting anchor, on brigs off the coast of California
A capstan song on a ship off the coast of Arabia
The adoption of capstan songs and sailors' "ditties" by locals of Tahiti and the Society Islands

1840s:

A rowing song ("row, billy, row") (American)
Reference to Black steamboat firemen singing on the Ohio
A stevedore song ("grog time") in New Orleans
Cotton-stowing songs in New Orleans and Mobile (x3)
Cotton "hoosiers" working (and singing) aboard a trans-Atlantic
    packet
[The "Stormy" chantey adapted (?) as a minstrel song]
[Popular shanties ("A Hundred Years Ago" and "Stormy") turning
    up in the South Pacific]
A song ("Cheerly Man") for hoisting guns from below in a ship
    from New York
Generic reference to "Cheerily Men" in Port Adelaide
Unloading cargo by means of a capstan in London, to a song
A basic anchor song ("Ho, O, heave O") brought back from a Mediterranean voyage in an American vessel
A walk-away shanty ("drunken sailor"), another shanty (unknown      
    style, "Nancy Fanana"), and a short-haul song ("O! hurrah my
    hearties O!") on an American whaling ship in the Pacific
Possible stanza-form chantey on a whaleship
Spoke windlass song on a whaleship
Capstan (or pump windlass?) chantey ("A Hundred Years Ago") on
    a steamship bound for Frisco
A halyard shanty ("grog time") on a brig in New York

Summary:
Through the 1840s, references to African-American work-songs for rowing and on steamboats are fewer (though they will continue to appear in later decades). The cotton-stowing songs make a big entrance here, and they seem to have strongly influenced shipboard work-songs.   Although I believe the cotton-stowing songs (as a class) must have originated with Black labourers, by this time Euro-American labourers had also joined the trade. If, as I have hypothesized, the body of songs of a style known as "chanties" originated with African-American practices, then a question at this point would be: Where and when did it become a shared practice among the different cultural groups? Was cotton-screwing, for example, a ground zero whence the practice was taken to ships? Or had the sharing already occurred earlier – e.g. aboard ships, initially? Whereas in the past I have thought that cotton-screwing songs must be analogous in form to halyard shanties, after a close look at the literature, I begin to have doubts whether that can be said with any certainty.

Interestingly, through the 1840s there are still not many references to what one might consider "classic" halliard shanties. For pulling songs, this decade gives us, with clarity, only the well-worn, old-fashioned "Cheerly"…. Along with "Grog time of day" (in two different forms) used for hauling to load cargo and…finally as a halliard chantey. I find the latter to be most significant, because it is (arguably) the one halliard song of the decade with something like the classic form. Still, it is shy of the most typical form (i.e. of later times).

Needless to say, all this material needs much more mulling over. But if I may state, prematurely, my surprise – that there is little evidence, by the end of the 1840s yet, to say that many of the chanties we now know (especially in the typical double-pull halyard form) had by then existed.