Thank you Twilight Fenrir (with your ragnarocky name).
Yes, I think the hairdressers have the upper hand. I shall henceforth think of this tool as a pinching iron. I suspect that the term and the tool are a bit generic: i.e., it could be used to 'pinch' other things than hair, even from the beginning.
First illustration below from: Hairstyles: a cultural history of fashions in hair from antiquity up to the present day; illustrated with objets d'art from the Schwarzkopf collection and international museums / Maria Jedding-Gesterline (1988). Copy taken from Google books snippet.
Second illustration (which seems to nail my tool pretty exactly) is from Modern Beauty Culture / Adelaide Smith, Reuben Rockwood (Prentice-Hall, 1934), p. 201; copy taken from HathiTrust.
There should also be a picture in A. Mallemont & Mark Campbell, The Techniques of Ladies' Hairdressing fo the 19th century : a compilation of original 19th century sources. (LACIS, 1996), p. 68. [Missing in Google snippet view.] Perhaps also in Victorian Houseware, Hardware, and Kitchenware : a pictorial archive / Ronald S. Barlow (Dover, 2001).
Pinching irons in use:
1. To curl hair in papers.
"Soft and fluffy gray hair often looks much better curled or crimped. If hot irons are used for this it will yellow the hair. ... It is therefore much better to do the hair up with papers or pins over night. If the iron is used at all, a very good effect may be obtained by weaving the hair on pins and covering the hair on each pin with tissue paper, before applying a pinching iron. Great care should be taken, also, not to have the iron too hot." -- Grace Murray, The fountain of youth: or, personal appearance and personal hygiene (New York, 1905).
"In curling the front hair or bangs, each lock may be twisted in a little papillotte, or a bit of paper, and then pinched with a 'pinching iron,' which is made to press between its two hemispheres a little flat curl." -- Beauty, its attainment and preservation (New York, 1892). [identical passage in The Delineator, July 1900]
"The secret of safe hair-dressing is never to pull the hair, never scorch, and always wrap a lock in paper before applying the iron. Common round curling-irons and frizzing-tongs may be safely used if thin Manilla paper is folded once around them. So in crimping: the hair may be done up on stout crimping pins held by slides, or braided in and out of a loop of thick cord, a bit of thin paper folded over the crimp, and the pinching-iron used with safety every day, providing the hair is not pulled too tight in braiding it The country method, where friseur's irons are unknown, is to lay the head on a table, and set a hot smoothing-iron on the woven lock -- an awkward but efficient process." -- Mrs. S. D. Powers, The ugly-girl papers, or hints for the toilet [from Harpers Bazar] (New York: Harper, 1874).
"Do the hair upon stout crimping pins, or braid it in and out of a loop of thick cord; fold a piece of thin paper over the crimp, and the pinching iron may be used in safety." -- Ward and Lock's Home Book; forming a companion volume to Mrs. Beeton's Book of Household Management (London, 1882?) [also published as Sylvia's Book of the Toilet : a lady's guide to dress and beauty.]
"HOW TO CARE FOR THE HAIR. I would advixe curling the bang by rolling it on a lead-pencil, putting papers over it, and then using a pinching iron, for this will not--unless one is absolutely careless--injure the hair in any way." -- Ladies' Home Journal v. 8 (May, 1891)
2. In baking.
"Mutton patties ... Line a dozen or more tartlet moulds with the trimmings of puff paste; fill them with the meat; cut out the cover, from the centre of which cut a piece the size of a wafer: pinch the borders all round with a pinching-iron." -- John Simpson, Simpson's Cookery, Improved and Modernised (London, 1834).
