The early days of 'chain mail'
The earliest Google Books match for chain mail that I could find is from Francis Grose, A Treatise on Ancient Armour and Weapons (1785):
Of mail there are two sorts, viz. chain and plate mail. Chain mail is formed by a number of iron rings, each ring having four others inserted into it, the whole exhibiting a kind of net work, with circular meshes, every ring separately rivetted; this kind of mail answers to that worn on the ancient breast-plates, whence they were denominated loricæ hammatæ, from the rings being hooked together.
Plate mail consisted of a number of small laminæ of metal, commonly iron, laid one over the other like the scales of a fish, and sewed down to a strong linen or leathern jacket, by thread passing through a small hole in each plate; this was exactly the form of the ancient lorica squammosa.
G. L. Way's 1796 translation of M. Le Grand, Fabliaux or Tales: Abridged from French Manuscripts of the XIIth and XIIIth Centuries (published in French in 1779) makes a similar distinction in a note to "Aucassin and Nicolette":
Mail armour, of which the hauberk is a species, and which derived its name from maille, a French word for mesh, was of two kinds: plate or scale mail (squamata vestis), and chain mail (hamata vestis). ... The engraving prefixed to 'The Order of Knighthood,' exhibits the scale mail, the plate and scale mail conjointly, and the chain mail.
And Samuel Meyrick, A Critical Inquiry Into Antient Armour, as it Existed in Europe, Particularly in Great Britain, from the Norman Conquest to the Reign of Charles II, volume 1 (1842) further distinguishes between single chain mail and double chain mail:
This armour is made by four rings joining a fifth, all of which are fastened with rivets. One row, indeed, on the hood, seems to double the number which would authorize the term double chain mail ; but as all the rest are as above described, the armour must be considered as the single chain mail. Yet mention is made in the Chronicle of Flanders of double chain mail, thus :
Un hauberk clavez de double maille.
"A hauberk of rivetted double mail."
On the correct English front, as part of a section on how to distinguish between mail and male, Henry Hopkins, A Key to Exercises in Orthography and Exercises in Composition on an Improved Plan, second edition (1844) has this sentence:
Some coats of mail were made of a great number of iron rings, and this was called chain-mail.
The campaign against 'chain mail'
Modern critics of "chain mail" seem to divide the world of armor into mail (chain mail) and "plate armor" (single-piece back-plate and breast-plate cuirasses), leaving the fishscale-style plate-mail forms (among others) out of consideration altogether.
One recent critic of "chain mail," Hugh Knight, The Play of the Axe: Medieval Pollaxe Combat (2009), offers this discussion of the topic in a glossary entry for mail:
Mail: A flexible form of armor composed of thousands of interlocking iron rings. These rings were either flat or (later) round cross section and were either riveted shut or made in one piece. Mail was worn under plate armor to protect the vulnerable joints that couldn't easily be protected by plate. Often mistakenly called "chain mail" today, but that term derives from a misunderstanding of medieval artwork and should not be used.
I don't know how to gauge the expertise of Hugh Knight (who may or may not have a twin brother named Hack); he doesn't provide any supporting documentation for his assertions in any part of the book reproduced online, and he seems to have published the book himself "through Lulu.com." Still, anyone who claims to be "Fechtmeister, Die Schlachtschule: The School of Battle" can't be dismissed out of hand just because he has a peculiarly apt name.
A much older criticism of chain mail appears in Francis Kelly & Randolph Schwabe, A Short History of Costume & Armour, Chiefly in England, 1066–1800 (1931):
In the Middle Ages and as long as armour was a living thing, the term mail denoted exclusively a defence of rings interlinked as in the modern woman's steel purse [cross reference omitted]. Only by a late poetical licence did the term come to be used of armour at large. "Chain-mail" is a mere modern pleonasm; "scale-mail" and still more "Plate mail," sheer nonsense. Meyrick's at one time generally accepted categories—"ringed," "mascled," "trelliced," "rustred," etc.—may henceforth be dismissed as fictions: so far as they were not pure inventions, they were based on a misconception of passages referring to so-called "chain" mail, whose antiquity and ubiquity are even now insufficiently recognized. Mail was in use in Western Europe centuries before the Crusades. So far from mail being imported from the East as a consequence of the Crusades, we are expressly told that it was the First Crusaders who brought it to Constantinople on their way to the East.
This view would seem to vindicate Hugh Knight and the other modern anti-chain-mailers. But it is not at all clear whether the Kelly & Schwabe view that various identified varieties of mail are either pleonasms (chain mail) or sheer nonsense (scale-mail and plate mail) has won the field in the brutal war over body-armor nomenclature. Interestingly, although in Wikipedia the search term "chain mail" redirects to the entry for "Mail (armour)," in a long and rancorous discussion on the subject of "ring mail/ring armour", both sides seem to find it convenient in the course of their discussion to be able to use the term "chain mail" to distinguish one type of mail/armor from others at issue in the controversy.
Kelly & Schwabe don't cite many authorities in support of their rejection of their Victorian predecessors' categorization of types of mail. This may indicate a casual attitude toward the utility of such citations, or it may indicate that they were the first revolutionaries of the anti-chain-mail school, so there was no one to cite on their behalf. But however you feel about chain mail (and I have very little feeling about it one way or the other), their denunciation of it as a false category comes across as far more mean-spirited and arrogant than it need be—even assuming that their analysis is superior to their predecessors'.
In the real world, people see pictures of chain mail that look different from pictures of scale mail and pictures of plate mail, and they think that the old categories make sense. Learning that these distinctions are (or may be) illusory is rather like being told that, though you might have grown up thinking that slate-colored juncos were one kind of bird, and Oregon juncos another, and white-winged juncos a third, and gray-headed juncos a fourth, in fact—even though each form looks quite different from the others—they're all just one kind of bird whose correct name is "dark-eyed junco." But it bears noticing that, having established the unifying idea of the dark-eyed junco, modern bird guides proceed to point out the slate-colored variant, the Oregon variant, the white-winged variant, and the gray-headed variant.
If the same residual value of the old categories is true in the case of mail/armor, it probably is true for the same reason: that the distinctions based on crude morphology are still useful when you encounter a specimen that seems to fall into one of the crude morphological categories. Experts in a field need to insist on the scientific rigor of their nomenclature; but for everyone else there is clarity, precision, and practical value in recognizing an Oregon junco or a chain-mail hauberk.