That will forever be the term.
This move alone almost surely ensures that 'tweet' as a reference to Twitter will _not_ live forever. Major publications, especially the NYT, control the progression of grammar and punctuation in the English language... almost exclusively. Their influence is undeniable with a great deal of modern grammar rules being able to be traced back to a single publication and/or editor's decision. Even while major newspaper circulation is down in the 21st century, these publications are still the gatekeepers to the rules regarding grammar and punctuation.
In terms of vocabulary, they will always take a cautious approach. They will choose words, terms or phrases that they think will stand the test of time and not become ambiguous. In this case, the editor feels that 'tweet' will not be synonymous with Twitter in the future and is acting accordingly. He is right.