I've found the only (???) reference to chantying in the 18th century, so far. It's basically something we've seen, but an earlier version with a much earlier date.

1784[1769]        Falconer, William. An Universal Dictionary of the Marine. New Edition. London: T. Cadell.

Falconer died 1769. He had sailed in British merchant ships from 1748 or '49 through the 1750s. I was not able to access the first, 1769 edition of this, but it seems to have had the same content as the 1784 (he was dead, and it doesn't look to have been revised). The passage is as follows.

From the entry on "WINDLASS":

As this machine it heaved about in a vertical direction, it is evident that the effort of an equal number of men acting upon it will be much more powerful than on the capftern; becaufe their whole weight and ftrength are applied more readily to the end of the lever employed to turn it about. Whereas, in the horizontal movement of the capfrern, the exertion of their force is considerably diminifhed. It requires, however, fome dexterity and addrefs to manage the handfpec to the greateft advantage; and to perform this the failors muft all rife at once upon the windlafs, and, fixing their bars therein, give a fudden jerk at the fame inftant, in which movement they are regulated by a fort of fong or howl pronounced by one of their number.
        
The moft dexterous managers of the handfpec in heaving at the windlafs are generally fuppofed the colliers, of Northumberland: and of all European mariners, the Dutch are certainly the moft aukward and fluggifh in this manoeuvre.

So, he is making the distinction between working the spoke windlass versus capstan. In his capstan entry, incidentally, no songs are mentioned. Capstan did not need coordination as, he explains, the spoke windlass did. He does not call it a "song" but rather a "SORT OF song or HOWL." Doubtful if this was any kind of "chanty" as we know them, however it was a similar practice. The question is whether it was anything more notable than a "yeo heave ho." Speaking of which...in his French dictionary, he later gives this entry:

UN, deux, troi, an exclamation, or fong, ufed by feamen when hauling the bowlines, the greateft eftbrt being made at the laft word. Englifh failors, in the fame manner, call out on this occafion,—haul-in—haul-two—haul-belay!

In a way, this is "negative testimony" suggesting that the French and English of that time did not have songs for hauling. True, that is not necessarily the case, and only the "bowlines" are discussed here. But I think it is reasonable to infer that if there were songs he would have mentioned them, no?