"Big Stick Diplomacy" was certainly not a term simply coined by historians to describe US policy under Roosevelt years after the fact, as it very clearly comes from Roosevelt himself, who defined his advice to "Speak softly and carry a big stick" as "the exercise of intelligent forethought and of decisive action sufficiently far in advance of any likely crisis".

As for whether anybody ever formally proposed to his nation that they formally adopt a policy of "Gunboat Diplomacy", I find i highly unlikely and have never come across any evidence. "Gunboat Diplomacy" is very simply a fancy specialized term for diplomacy through intimidation, which is a tactic as old as politics itself.

The popularization of the term in historical circles mainly stems from the work of James Cable who, in 1981, after extensively studying the naval history of great powers such as the US and UK over the preceding centuries defined it as "the use or threat of limited naval force, otherwise than as an act of war, in order to secure advantage or to avert loss, either in the furtherance of an international dispute or else against foreign nationals within the territory or the jurisdiction of their own state."

Not to take away anything from Cable, but the concept of sending a show of force to impress or demoralize a foreign people or their leader was not a revolutionary idea. While many do agree that the term "Gunboat Diplomacy" is an acceptable way of describing intimidation tactics primarily involving naval forces being sent to foreign waters, the idea that seems to be implanted in the heads of many students nowadays that this was a phenomena solely limited to the imperial powers of the 19th and early 20th centuries is absurd.

Sources and Further Reading:

Cable, James (1981). Gunboat Diplomacy. Basingstoke: Palgrave Macmillan.

Roosevelt, Theodore (1913), Theodore Roosevelt: An Autobiography, The Macmillan Press Company