The author W. Clark Russell had experience as a sailor (before 1865), but he was also very much a "man of letters." His writings that mention chanties, beginning in 1883, describe them/quote them / title them largely based on his reading, rather than what experience he may have had. In other words, though we can envision his first-hand experience, the framework of his discussion is prior authors'.

1888[June]        Russell, W. Clark. "The Old Naval Song." Longman's Magazine Vol 12 (June 1888): 180-191.

If most of the forecastle melodies still current at sea be not the composition of Yankees, the words, at all events, are sufficiently tinctured by American sentiment to render my conjecture plausible. The titles of many of these working songs have a strong flavour of Boston and New York about them. 'Across the Western Ocean'; 'The Plains of Mexico'; 'Run, let the Bulljine, run !' 'Bound to the Rio Grande '; these and many more which I cannot immediately recollect betray to my mind a transatlantic inspiration. 'Heave to the Girls'; 'Cheerly, Men'; 'A dandy ship and a dandy crew'; 'Tally hi ho! You know'; ' Hurrah! hurrah! my hearty bullies'; and scores more of a like kind, all of them working songs never to be heard off the decks of a ship, are racy in air and words of the soil of the States.

Despite my rather grandiose comment on the nature of Russell's writing, I can't say much based off the short phrase, except that the wording "bound to the" had previously appeared in the 1869 Chambers's, in Symondson's account, and in the Davis/Tozer collection. Which might mean nothing. It is possible (but, again, maybe meaningless) that Russell had peeked at Davis/Tozer's 1887 work, because in Russell's writings prior to 1887, though he mention several chanties he does not mention this one.