In Longman's Magazine Vol 12 (June 1888) there is an article "Old Naval Songs." The author, W. Clark Russell, sees fit to bring in chanties, although he familiarity with them is questionable. Several that he mentions are clearly pulled from Dana's TWO YEARS, but he passes of the titles as if they were songs that were part of his experience.

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THERE are two kinds of sea-songs: those which are sung at concerts and in drawing-rooms, and sometimes, but not very often, at sea, and those which are never heard off shipboard. The latter have obtained in this age the name of ' chanty,' a term which I do not recollect ever having heard when I was following the life. It is obviously manufactured out of the French verb, and there is a 'longshore twang about it which cannot but sound disagreeably to the elderly nautical ear.
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Hmm. Never heard the term 'chanty'? Possible. When did he sail?

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This sort of song is designed to lighten and assist the sailor's toil. It is an air that enables a number of men pulling upon a rope to regulate their combined exertions. It is also a song for sailors to sing as they tramp round a capstan and heave upon a windlass. Of the melodies of many of them it is difficult to trace the paternity. Some are so engaging that they might well be regarded as the compositions of musicians of genius, who wrote them with little suspicion of the final uses to which they would be put. Why their destination, having been sung perhaps at the harpsichord and the guitar by ladies and gentlemen, should be the forecastle; why, being appropriated by the sailor they should be so peculiarly his, that no one else ever dreams of singing them, there is no use in attempting to guess.
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Forecastle? He's talking about chanties, right?

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The reader will not require me to tell him that the marine working songs are to be heard only in the Merchant Service. In a ship of war the uproar caused by the hoarse bawling of half-a-dozen gangs of men scattered about the decks would be intolerable, nor could the working song be of service to the blue-jackets, who are quite numerous enough to manage without it. It was always so, indeed ; a frigate getting under way would flash into canvas in a breath ; sails were sheeted home, yards hoisted, jibs and staysails run up, and the anchor tripped, as though the complicated mechanism were influenced by a single controlling power producing simultaneously a hundred different effects. There were men enough to do everything, and all at once ; but the ship's company of the merchantman were always too few for her. A mercantile sailor is expected to do the work of two, and, at a pinch, of three and even four. When one job is done he has to spring to another. There are 'stations' indeed in such manoeuvres as tacking or wearing; but when, for instance, it comes to shortening sail in a hurry, or when the necessity arises for a sudden call for all hands, the merchant sailor lays hold of the first rope it is necessary to drag on, and when he has ' belayed' it, he is expected to fling himself upon the next rope that has to be pulled. Here we have the secret of the usefulness of the working song. Let the words be what they will, the melody animates the seaman with spirit and he pulls with a will; it helps him to keep time too, so that not so much as an ounce of the united weight of the hauling and bawling fellows misses of its use on the tackle they drag at. I have known seamen at work on some job that required a deal of heavy and sustained pulling, to labour as if all heart had gone out of them whilst one of the gang tried song after song ; the mate meanwhile standing by and encouraging them with the familiar official rhetoric; till on a sudden an air has been struck up that acted as if by magic. The men not only found their own strength, every fellow became as good as two. This, I believe, will be the experience of most merchant sailors.
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OK, so more confirmation of the lack of chanties in navy ships, and the reasons why.

Next, types of chanties:

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There are tunes to fit every kind of work on board ship; short cheerful melodies for jobs soon accomplished, over which a captain would not allow time to be wasted in singing (for I am bound to say that the disposition of a sailor is to make a very great deal of singing go to the smallest possible amount of pulling), such as hauling out a bowline, mastheading one of the lighter yards, or boarding a tack. Other working choruses, again, are as long as a ship's cable. These are sung at the capstan or at the windlass, when the intervals between the starting of the solo and the coming in of the chorus do not hinder the work an instant.
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Next he states he could not fins mention of chanties in older British accounts, concluding they may be American in origin:

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It would be interesting to know when and by whom the working song was first introduced into the British Merchant Service. In old books of voyages no reference whatever is made to it. There is not a sentence in the collections from Hakluyt down to Burney to indicate that when the early sailors pushed at handspikes or dragged upon the rigging they animated their labours with songs and choruses. I have some acquaintance with the volumes of Shelvocke, Funnell, and other marine writers of the last century, but though many of them, such as Ringrose, Dumpier, Cooke, Snelgrave, and particularly Woodes Rogers, enter very closely into the details of the shipboard work of their time, they are to a man silent on this question of singing. It is for this reason that I would attribute the origin of the practice to the Americans.
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If most of the forecastle melodies still current at sea be not the composition of Yankees, the words, at all events, are sufficiently tinctured by American sentiment to render my conjecture plausible. The titles of many of these working songs have a strong flavour of Boston and New York about them. 'Across the Western Ocean'; 'The Plains of Mexico'; 'Run, let the Bulljine, run !' 'Bound to the Rio Grande '; these and many more which I cannot immediately recollect betray to my mind a transatlantic inspiration. 'Heave to the Girls'; 'Cheerly, Men'; 'A dandy ship and a dandy crew'; 'Tally hi ho! You know'; ' Hurrah! hurrah! my hearty bullies'; and scores more of a like kind, all of them working songs never to be heard off the decks of a ship, are racy in air and words of the soil of the States.
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'Heave to the Girls', 'Tally hi ho! You know', 'Hurrah! hurrah! my hearty bullies'
'Cheerly, Men', 'A dandy ship and a dandy crew' -- clearly come from Dana.

'Bound to the Rio Grande' could be his observation. But that exact wording appeared also in Chambers's Journal, 1869.

'Across the Western Ocean' - This exact phrase has not turned up yet in this thread.

'The Plains of Mexico' -- was there prior.

'Run, let the Bulljine, run!' -- I'm seeing this for the first time.


Then he moves onto sea songs, emphasizing the improvised, incidental, and 'doggerel' nature of chanty verses:

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The other kind of songs—the songs of Charles and Thomas Dibdin, Shield, Arnold, Arne, Boyce, &c., are of a very different order. The working song is often at best but little more than unintelligible doggerel. It is the sailor's trick to improvise as he goes along, and rhyme and reason are entirely subordinate to the obligation of shouting out something. But the sea-song, as landsmen understand the term, is accepted as a composition of meaning and even of poetry...
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