A long time ago, I heard Martin Wyndham Reid (Reed?) O Waly Waly at a house concert. I'm not going to put quotes around song titles here. Makes the typing too picky. Here are the lyrics:

When cockleshells turn silver bells,
And mussels grow on ev’ry tree;
When blooms a rose ’mid frost and snows
Then will my false love prove true to me.

Chorus (repeated after each verse):
O waly, waly, ah, love it is bonnie
A little while when first it’s new.
But love grows old and waxes cold
And fades away like the morning dew.

Ah, had I wist, before I kissed
That love could be so ill to win;
I’d locked my heart in a box of oak
And pinned it shut with a silver pin.
O waly, waly...
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All well and good. It's called Waly, Waly and it has the words 'Waly, waly' in it.

Enter the old ballad collectors. They didn't seem to be interested in songs and singing; they were interested in language - lyrics, plots, symbols, rhymes. For them, though two songs had completely different tunes but the same topics, they were the same thing. When they encountered 'The Water is Wide' and heard many of the same words, they dubbed it Waly,Waly too. Never mind that the words 'waly, waly' never occurred and the tune was completely different. Also Waly Waly seems older and more highbrow. Few have ever heard of it, while thousands have sung the Water is Wide.

As far as I'm concerned, the song that starts 'The water is wide' is The Water is Wide. Martin's song is Waly, Waly.