Probably they all use the same tune, and will be arrangements of the usual form of the song that turns up on commercial recordings (though in some cases with verses added or subtracted, and words changed or moved around) which can be traced almost invariably to a set published early in the 20th century.

The Water is Wide, as most people know it, was found by Cecil Sharp in Somerset in the early years of the 20th century (the text is a collation from more than one singer); he published it between 1904 and 1906 in the UK, and in 1916 in the USA. The commonly-recorded form (popularised in the American folk revival by Pete Seeger, who got it from his sister Peggy, who seems to have got it from one of Sharp's books) is not Irish at all, but English. Seeger added a verse to it himself.

Part of the confusion over the song is that some early collectors, noticing that it shared verses with Waly Waly decided that it "must" be a "version" of that song; a misapprehension which has stuck, and is regularly repeated to this day. Sharp didn't actually make that claim, though he published it as Oh Waly Waly, commenting "The words are so closely allied to the well-known Scottish ballad ... that I have published [it] under the same title."

In truth only a few verses are shared (and not the "water is wide" verse); and many of these turn up in many other songs. For the uninitiated, such things are generally called "floating" verses, and are a commonplace of traditional song, capable of leading the unwary into guessing at all manner of connections which often turn out to be mere illusion. Whole songs are composed of such verses, and regularly lead folk into the error of assuming things like "Waly Waly and The Water is Wide are [versions of] the same song". That is a gross over-simplification. They share material, but are distinct, separate pieces.

This being the case, it is hard for us to be sure what the "complete" version you mention might actually be. The song (and others like it, related and unrelated) has been discussed here at considerable length in the past, though, and you might like to have a look at some of the earlier threads, to which links have now been added at the head of this page.

Once you've found the lyric that you know as the "complete" one, point us to it (don't bother to quote it here; it will almost certainly have been posted several times already) and people will then be able to tell you who has recorded it. If, as I suspect, you mean the Sharp collation (the "DigiTrad" example shown above as THE WATER IS WIDE derives from it, though the words have been changed around a bit), then people have been recording arrangements of it for a long while -some have already been mentioned- though it is only in recent years that the marketing boom in "Celtic" music has led people to imagine that it is an Irish song. There are related songs in Irish tradition, as in those of all other countries where English is spoken; but they have their own tunes, and rarely include the "water is wide" verse.