Aunt Mai's 1894 Annual, online and readable. Do a search-this-page for "Mai's." They've misspelled Mrs. Steinthal's name as author, but it doesn't matter. Smyrna rugs (called Door Mats) on page 119. Also the Japanese curtains etc. etc.
And the key to the whole question is...LATCH HOOKING. Yep. No knitting, no weaving...the PNEU programme specifically recommended this particular set of instructions, so the official CM "Smyrna rugs" were short pieces of yarn pulled through canvas with a crochet hook. That's it, that's all.
As for materials, Aunt Mai recommends the following for one door-mat: 1/2 yard of canvas, a yard wide; 1 1/2 pound of "thrums, or the leavings of carpets"; 8 lengths of wood, 7 or 8 inches long and 1/4 inch wide ("The lid of a cigar-box furnishes the best wood"); a crochet-hook; pair of scissors.
Oh, what the hey, might as well write the whole thing:
"Let the pupil wind each piece of wool separately round the wood, and cut through one end, leaving each piece of wool 2 1/2 inches long. Place the canvas on a table and draw the chief lines of the pattern in pen and ink, or with a brush and sepia. Choose the colour to begin with; put the two ends of the piece of wool together; put the crochet hook through the first hole, draw the wool through, loop it, and pull it to make it quite firm. The wool is put in and drawn tight, exactly as the fringe used to be put on to antimacassars. It is the best plan to begin with the border, and then decide what pattern will look best in the centre, and last of all, to choose the background....Dark blue is effective, also a very dark terra cotta.
"Cross-stitch patterns can easily be copied, but after one or two attempts the children find a keen pleasure in inventing thei rown designs.
"When the mat is finished, brush the back with glue and a little flour added, which stiffens the rug and prevents any wool getting loose. Then seam on a strong piece of canvas, and a strong, beautiful mat is produced, which will last for years."
There you go.