Is this a bit of journalistic license?
The statement literally means "Twitter is the only way for world leaders to communicate with one another using Twitter," so it's true but fairly meaningless. If you read between the lines and determine that the speaker probably meant to say "Twitter is the only channel for diplomatic communication available to some leaders" then no, it's utter nonsense. Diplomatic communication is generally conducted in private, not in public.
Are there at least notable specific cases of countries without diplomatic relations where they have telegraphed positions, postures or information to one another via tweets?
There may be, but this is a very different question. Public statements are not "a diplomatic channel of communication," and the fact that someone may have made such statements via Twitter does not imply that Twitter was the only "instrument" for making such statements.
In response to a comment from uhoh,
"The statement literally means 'Twitter is the only way for world leaders...'" No it literally says something different than that. I have it in bold font there and I'll do that here as well: " ...who have no other diplomatic channel to speak to one another get the word out to one another..." We don't need even more manufactured outrage here. Also, can you support why public statements can never be a diplomatic channel of communication or adjust your last sentence so that it's not so absolute?
Ok, yes, it is perhaps imprecise of me to paraphrase something while using the phrase "literally says." It literally says "We're talking about the instrument with which world leaders who have no other diplomatic channel to speak to one another get the word out to one another across Twitter." But "world leaders who have no other diplomatic channel to speak to one another" is only a subset of people for whom Twitter is "the instrument with [they] get the word out to one another across Twitter." The full set of such people is all Twitter users. Twitter is the the only instrument with which anyone can "get the word out across Twitter" just as NBC News is the only instrument with which anyone can get the word out through NBC News. It is a true statement of trivial utility.
Now of course the phrase "using Twitter" at the end of the statement can be written off to the imprecision that often arises in speech. Discounting that phrase makes the statement somewhat more meaningful, which is why I also note that the set of "world leaders who have no other diplomatic channel to speak to one another" is the empty set, because everyone has access to telephone and e-mail, and world leaders furthermore have access to diplomatic intermediaries from other countries with mutual diplomatic relations, whether with formal recognition as a protecting power or otherwise.
As to the last sentence, public statements are a public channel of communication. Such statements are of course a significant element in diplomacy, but they are not a diplomatic channel of communication. The phrase "diplomatic channel" typically refers to official communication through diplomats. See, for example: