But many sailors themselves DID make the distinction - the shanty or chantey (take your choice on the spelling) was a working song, fitted to the rhythm of the task and lasting as long as the job lasted. Forebitters or fo'c'sle songs (again, take your choice) were sung for pleasure, shanties - at least on working sailing ships, were never sung recreationally.
Certainly the British practice was that the shanty was a merchant marine thing - singing at work in the Royal Navy was a flogging offence. Shantying made up for the short-handed crews as sail tried to compete with steam - a good shantyman was worth ten extra hands on the rope.
There is no "right" set of words for a shanty. Each shantyman had his own sets, would make up verses about events or characters on board, and if the job went on after he'd finished his "normal" set of verses, he had to keep the song going, by bringing in verses from other shanties or making them up. Texts in books are just the way the song was sung once, by one particular shantyman, who may never have sung it the same way before or since.
Shanties are many things, but "exactly" they aint.

Andrew