I'll try and offer a few facts about the early use of the term "back saw."
The first use recorded in the OED is in Edward H. Knight "The practical dictionary of mechanics" 1st edition, 1874–1884 (4 vols.) London: Cassell, dated to 1877
Back-saw, a saw whose web is stiffened by a metallic back of greater substance; as, a tenon saw.
I have found one earlier use recorded - the William Marples catalogue of 1864, reproduced (rather small and fuzzy, but readable) in Goodman's "History of Woodworking Tools" on page 141, illustrates a
C.S. [ie Cast Steel] Back-Saw, pattern no 1213.
I can't find anything else earlier than the Holtzapffel cited by CheshireChappie - which ought to be in the OED.
Smith's Key (the early standardised common catalogue of Sheffield Toolmakers) does not use the term in the 1815 edition. It does in the 1871 edition, in both the illustrations and in the price list.
The 1938 Marples catalogue uses the term Back Saw. So does the virtually identical catalogue of Turner, Naylor and Company (ie the catalogue of tools branded with the I. Sorby "Punch" mark).
After all those facts, one opinion: the term 'back saw' is quite a useful one if you want to distinguish saws by their construction rather than their use.
EDIT: Ok, one older example, from famous social reformer Robert Owen, writing in The New Moral World, in
1839, published in Leeds, listing the presents given to the Hampshire Community:
Thank you Google.