They probably mean tweeting something like "At 11:45 EDT, the United States will be launching a tactical missile strike against Iran's Natanz and Karaj nuclear reactors, unless the Ayatollah agrees to permanently cease their operations", not "Fatty Kim Jong-Un is a loser!"
I doubt it would actually start a war, but it could definitely cause problems, especially if they believe the tweet may have actually been a "testing of the waters" rather than a result of a compromised account.
That shared understanding has never existed in history. Intelligence and diplomacy has always been part of warcraft. The Spanish-American War and Pearl Harbor are two notable cases of war triggered by communications.
I doubt it would actually start a war, but it could definitely cause problems, especially if they believe the tweet may have actually been a "testing of the waters" rather than a result of a compromised account.