Stainer’s Christmas carol lyrics were published in 1912 (see advertisement image on this page; ad might be offline by now).
The lyrics were reprinted (without crediting Stainer) in Carols Old and Carols New, by Charles L. Hutchins (Boston, Massachusetts: The Parish Choir, 1916), number 498.
It seems likely C. L. Stainer
might be Charles Lewis Stainer (1871–1947), given the Stainer family’s interest in carols: Charles’ father John Stainer was co-author of Christmas Carols, New and Old in 1871.
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