I tracked it down to this French journal Journal de mathématiques élémentaires (1896) pp. 22-23 which was edited by Gohierre de Longchamps.
Nous avons reçu de. M. Plackowo (à Tokarewka, gouvernement de Tamboff) la question suivante. Nous lui avons envoyé la solution qu'il désirait avoir de cette règle curieuse, practiquée par les paysans russe, règle qui évite la conaissance de la table de Pythagore. Cette démonstration est basée sur le système binaire de numération. Mais peut-être un de nos correspondants trouvera-t-il une solution plus simple.
We received from. Mr. Plackowo (in Tokarewka, Tamboff government) the following question. We sent him the solution he wanted of this curious method, practiced by Russian peasants, a method which avoids knowledge of the multiplication table. This demonstration is based on the binary number system. But perhaps one of our correspondents will find a simpler solution.
I was able to find the reference to this because I am fortunate enough to have access to JSTOR. Google brought up this article:
Archibald, R. C. (1918). Undergraduate Mathematics Clubs. The American Mathematical Monthly, 25(3) 132-142
which contains a poor reference to the original publication:
Tamboff is known today as as Tambov while Tokarewka appears to be known as Tokaryovka.
The Bryan reference is here. The Bowen reference has more lousy referencing:
Cosmos appears to be this, but I didn't find it online. It was only when I went looking for the E. Czuber reference that I found the proper reference in a German book (W. Ahrens (1918) "Altes und Neues aus der Unterhaltungsmathematik", p. 83) on Google Books ("dort Hinweis auf Journ. de mathém. élém. 1896, p. 22, 23, 37"), also quoting a comment "Wir möchten einen leisen Zweifel dazu äußern, daß russische Bauern wirklich auf diese Weise multiplizieren" (translation: we may express a slight doubt that Russian peasants really multiply this way") from the E. Czuber source. (I couldn't find the Czuber article online, though.)
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