Traditional maritime music is a broad category that is often cross pollinated with other folk music of the times. For this explanation we're just going to divide it into three groups.
First we have actual sea shanties. These are songs that were sung by sailors while they were doing work on ships ( or next to them for the purpose of loading). These were not songs that were sung for enjoyment but to keep everyone in time, they weren't sung off ships and in fact some sailors found it bad luck to sing them off of ships. All these songs would be sung acapella, many of them featuring call and response. The shanties are then further divided depending on the length and nature of the task. So we have shanties for hauling, heaving, the capstan, the windlass, the pumps and so on. As shanties could vary in length depending on how long it took to complete the task, the verses tended to be modular. This meant that verses from one shanty may show up in another shanty depending on if you need to extend the song. It also meant that there was no set order for how the verses should go, as the song wasn't really about telling the story it was about maximizing efficiency on the ship, and keeping everyone in time. These songs were also developed organically from sailors on maritime vessels and would be shared through an oral process. Often they would be written down by other people.
Second, we have songs that are sung off-watch. These songs are called forebitters or fo'c'sle songs. These are songs that sailors would sing for their own enjoyment. Many of these songs would also be popular songs of the day that sailors would learn when they were in port. Unlike shanty's these songs could often have musical accompaniment.
Finally, we have just generic sea songs. These are songs that were written about various maritime subjects but they were written and performed on land for audiences there.
Now other songs you specifically brought up, Old Maui was most definitely sung on ships. According to Gale Huntington in his book "Songs the Whaleman Sang" there is a version called "Rolling Down to Old Mohee" from a journal that was written aboard the vessel Atkins Adams in 1858. Starting in the 1820s Maui, and the Hawaiian islands as a whole, were utilized as home bases for Yankee whaling fleets. Now this song was most definitely a forebitter sung while the sailors were relaxing. There is a narrative structure in the lyrics that necessitates a certain order as well as the general rhythm of the song not being conducive to work. The most popular version of this song uses the melody that originates from "The Miller of Dee". That song originated in the 1762 ballad opera "Love in the Village" by Isaac Bickerstaffe and became popular in Northwest England.
The Wellermen is a sea ballad that comes from New Zealand. Now the lyrics tell a story as well as the different melodies of the song this isn't really conducive to being sung as a shanty. If we look at the subject matter as well as the context it helps give us an idea. According to "Folksongs of New Zealand" by Neil Colquhuin, New Zealand whalers practice shore whaling ( also known as bay whaling). This was something that was common in New England in the early colonial days. First they would go out on small boats, kill the whale and then bring it back to shore for processing. These whalers were not paid wages and therefore were paid "stake". This would include ready-made goods such as tobacco, liquor or clothing. These goods of course would be supplied to their workstations via supply ships run by the Weller Brothers. The trading agents on these ships were referred to as "Wellermen". Now the song does describe tall ship whaling, going after an elusive right whale, but this seems to be a fictional creation as opposed to something drawn from their own experience. No we can't say for sure this was never sung on a ship as say a forebitter. But what we do know is the first time it was collected was in the 1960s by Neil Colquhuin in New Zealand. The song does not show up in collections from other maritime nations. So even if it was sung on ships as a forebitter in that area, the song might not have caught on with crews from other parts of the world.
I do want to end this with a little side note. That sometimes The crew of a whaling ship may be forbidden to sing on their off time depending on the captain who was king. Of course it varied from ship to ship. I highly recommend "Leviathan: The History of Whaling in America" by Eric Jay Dolin. Of course it's focus is the development of that industry in America, not really focusing on the music aboard these ships. That said it is a very well researched book that can fill you in not just about the life of the whalers themselves but their place in the world economy at the time.
Also I will summon u/JBSongman who has done much study on maritime music.
I answered a similar question here.
I quoted Robert Adams' 1879 memoir, On Board the Rocket , where he described pretty exactly how shanties would be used. And I'll quote him again:
THE songs of the sea offer a field for research, and one who could trace the origin and use of some of them would doubtless discover interesting, romantic histories. No information can be obtained from sailors themselves on this point. No one knows who their favorite "Reuben Ranzo" was, or whether "Johnny Boker" ever did what he is so often requested to "do," nor can any one say more concerning the virtues and vices of "Sally Brown" than is declared in song.
Sailors' songs may be divided into two classes. First, are the sentimental songs sung in the forecastle, or on the deck in the leisure hours of the dog-watch, when the crew assemble around the fore-hatch to indulge in yarns and music. Dibdin's songs, which the orthodox sailor of the last half century was supposed to adhere to as closely as the Scotch Presbyterian to his Psalter, are falling into disuse, and the negro melodies and the popular shore songs of the day are now most frequently heard. The second class of songs is used at work, and they form so interesting a feature of life at sea, that a sketch of that life would be incomplete without some allusion to them. These working songs may be divided into three sets:
First, those used where a few strong pulls are needed, as in boarding a tack, hauling aft a sheet, or tautening a weather-brace. "Haul the Bowline," is a favorite for this purpose. The shantyman, as the solo singer is called, standing up "beforehand," as high above the rest of the crew as he can reach, sings with as many quirks, variations and quavers as his ingenuity and ability can attempt, "Haul the bow-line, Kitty is my darling;" then all hands join in the chorus, "Haul the bow-line, the bow-line haul," shouting the last word with great energy and suiting action to it by a combined pull, which must once be witnessed by one who desires an exemplification of "a long pull, a strong pull, and a pull altogether." This seldom fails to make the ropes "come home."...
The second kind of shanty, says Adams, is used for a long series of moderate pulls, like for the topsail halyards. The third kind is for a continuous pull or push where it's useful to have the crew keep in step: like using the capstan to raise the anchor.
Dibdin would be Charles Dibdin (1745-1815) an extraordinarily prolific English composer, singer, actor, playwright, and novelist. It's hard to say someone who knocked out over 1,000 works in his lifetime specialized in sailors' songs, but he wrote many. You likely haven't heard one in a pub recently....
THE WANDERING SAILOR.
Undaunted braves the stormy seas, To find, at last, content and ease: In hopes, when to land danger's o'er, To anchor on his native shore. When winds blow hard, and mountains roll, And thunders shake from pole to pole; Tho' dreadful wares surrounding foam, Still flatt'ring fancy wafts him home; In hopes, when toil and danger's o'er, To anchor on his native shore. When round the bowl, the jovial crew, The early scenes of youth renew; Tho' each his fav'rite fair will boast, This is the universal toast—
chorus THE wand'ring sailor ploughs the main, A competence in life to gain
Not to be judgmental, here, but just as Moore's "Sublime Was The Voice With Which Liberty Spoke" hasn't been revived, it is safe to say that few musicians would be singing this now, and it's easy agree with Adams: even by 1879 Dibdin's very 18th c. poetic style would have given way to more straightforward popular songs, like "Old Mohee" , first mentioned ( though not transcribed) in a whaler's journal in 1858.
Rolling down to Old Maui, me boys Rolling down to Old Maui We're homeward bound from the Arctic Ground Rolling down to Old Maui
Dibdin, Charles. Dibdin's Museum Evans' Early American Imprints
Adams, Robert. ( 1879) [On Board the Rocket, ch.12] (https://www.gutenberg.org/files/64171/64171-h/64171-h.htm#CHAPTER_XII)
According to Gerry Smyth in Sailor Song: The Shanties and Ballads of the High Seas, sea shanties were quite popular on merchant ships and whalers in the late 1800s.
See shanties were strictly in a call-and-response format, and were sung to keep a rhythm to which the sailors should be working (the type of hard labor they were doing on the ships required groups of men to be doing the same thing at the same time).
However, the song you gave is an example, The Wellerman, is not actually a sea shanty as would have been understood at the time—as it is not sung in a call-and-response format. Smythe makes the distinction between sea shanties—which are essentially work songs—and the sort of communal ballads the sailors might sing onshore or in a pub. Or songs that are simply nautical-themed folk songs.
Another point made in this book was that many sea shanties and maritime songs that managed to stay in the public consciousness have been changed. Bawdy lyrics would have gotten censored/filtered by non-sailors' Victorian sensibilities, and of course the English language as a whole has changed quite a bit from the late 1800s to today. I don't recall if the other examples of songs you gave were discussed by Smythe.
You've gotten some good answers already, OP but I'd also like to point you to this amazing 4-part answer written by /u/dgbd to a similar question I asked last year!
For more about the work song genre, listen to Episode 169 of the AskHistorians Podcast feat. /u/Kelpie-cat