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In the West, the earliest publication of the Lily appears to be in "Images a Plier", by M.R. Chasles, which was listed by David Petty in his posting. This "book" was published in parts in France in the 1890s. Robert Harbin mentions this book in the bibliography of "Paper Magic", although he mis-spells the name of the author as "Charles".
Robert Harbin may have taken the name of his book from "Paper Magic" "More Paper Magic" by Will Blythe (also a magician), which were published in London in 1920 and 1923 and which both contain a great amount of paper folding. The Lily is not included in either book, but the Jumping Frog is incuded in "More Paper Magic".
Similarly, the Jumping Fog, but not the Lily is included in "Fun with Paper folding" by W.D. Murray and F.J. Rigney, published in New York in 1928
The traditional model of the Lily appears in Margaret Campbell's "Paper Toy Making", which was printed by Sir Isaac Pitman & Sons Ltd of London in 1937. Mrs Campbell unambiguously calls it a Lily. Curiously, while Margaret Campbell includes the Frog Base later in her book, she doesn't use it for the Lily, but folds it as hoc. Later in her book, Margaret Campbell includes the Frog Base under the name of the "Third Foundation". In Paper Magic (1956), Robert Harbin renamed the Third Foundation as "Basic Fold Three". From it he folded several varieties of frogs and other models, but, curiously he did not include a Lily, despite the fact that he had been inspired to take up paperfolding in 1953 when he came across a copy of "Paper Toy Making"
It was only later, in 1961,that Sam Randlett, who drew much of his basics of folding from Robert Harbin, renamed the "Third Foundation" or "Basic Fold Three" as "the Frog Base" in his book, "The Art of Origami" (1961). He gave names to the bases which he chose for his book on the principle that each of the bases he named should be named after a common model folded from it.
The Jumping Frog was introduced into Europe by the same Japanese magicians who introduced the Flapping Bird, apparently between about 1870 and 1880, when Japanese magicians began touring European and American theatres and exhibitions. The models also introduced Westerners to the Bird Base and the Frog bases. The Flapping Bird was included in the Boy's Own Paper in 1806 and the Jumping Frog soon after that. The Bird Base and the Frog Base introduced a completely new way of folding to Europe. The Bird Base, especially inspired the Spanish philosopher, Miguel Unamuno to start a new trend of folding in Spain and later in Argentina, but not, it appears, to Northern Europe.
When "Paper Toy Making" was published, Margaret Cambell, who was a Scotswoman, was living in South Africa and she wrote the book for her grandchildren. She had married a diplomat and spent some years in the East, including Japan. "Paper Roy Making" contains not only models already known in Europe, but also a number of models apparently known there and apparently learnt by her in Japan, or elsewhere in the East. I had discovered that the Jumping Fog probably didn't come to the West with the Japanese magicians. It was likely to be one of the folds brought by Japanese educators to one of the International Expositions held at Paris in the 1880s, where there was a display of children's folding
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