Mortartube, Tony Dalton's Maritime Colloquialisms...your source... sounds as if he might just be an ex-sailor who happens to have gathered together a few ideas he's heard on his travels. Michael Quinion's Worldwidewords...TCL's source... on the other hand is the work of a man known to be an etymologist and lexicographer; that is a genuine expert on language and its forms/history.
The Oxford English Dictionary - the 'bible' of English etymology - does not even list the 'tow-rag' form.
As you yourself say, much depends on one's "online source" and there is no doubt at all as to which of the two under consideration here is the more authoritative.
Re the actual question, people need to be sure they know which words/phrases actually are Americanisms. Recently, here on AnswerBank, someone was complaining about all the ise/ize words...such as prioritize...creeping into British English, all of them supposedly 'hated Americanisms'. The problem is, not all such words are! 'Diarise', for example, is British in origin and there are lots of others.
Surely, if a word/phrase fulfils a function, it is worth having wherever it comes from and many American word-coinages are brilliantly creative. Just think what a poverty-stricken language British English would be if we banished all words without roots in Old English! No kiosk...Persian...no telephone...Greek...no sausages...French...and so on.
Let 'em all come and - if you don't like 'em, don't use 'em!