> Also where does he get "Water-sprite"...?

As you say, it's "mermaiden" or "mermaid" in all three versions in Child. "Water-sprite" looks to me more like a scholar's term than one that would appear in a traditional song -- I notice it is used by Bayard and Harbison Parker in their articles, for example. But in any case, I'm not sure that the particular label put on a supernatural being is so important. It's the behaviour attributed to them and their function in the story that matter.

If we think of a mermaid as a sea creature, half woman half fish holding a comb and a mirror, who entices sailors to their doom with her singing, then the supernatural being in Child 42 is clearly something rather different. She lives in fresh water and seems to change from completely human to completely fish form in the course of the story, rather than being half and half. But there are plenty of freshwater mermaids in English and Lowland Scottish folkore. In "The Lore of the Land: A Guide to England's Legends" (2005), Jennifer Westwood and Jacqueline Simpson write: "Mainly [...] mermaids lived in pits and pools. Although sharing the name 'mermaid' with the half-fish creature of the bestiaries and heraldry, they represent native tradition rather than learned lore. The 'mer' of mermaid is Old English *mere*, a pool [...]"