Malcolm, thanks for pulling the versions together.

Nigel, the "Scottish Students Song Book" is a good source for the period. A number of the songs were popular on both sides of the Atlantic. "Over the Sea ..." was included in the ed. which I have. Like yours, it must be the 6th of 1897 since Stevenson's version was not published until 1896.
A different song, "Over the Water to Charlie," is found in the U. S. song book, "Franklin Square Song Collection (1)" of 1881 (1898).

The "Skye Boat Song" seems to have taken a few years to become popular in America (sheet music in England, 1884. Stevenson's appeared in "Songs of Travel and Other Verses," 1896). No copies of sheet music listed in American Memory or Levy.
The "Skye Boat Song," Boulton- Lawson, ("Jacobite") is printed in the 1903 edition of "Folk Songs and Other Songs for Children," an Oliver Ditson book. Did it appear earlier in any of the popular American song books?