I'm not a doctor. Pretty much the furthest thing from it. I am a cuemaker and cultural anthropologist / historian. The caduceus is often used to symbolize the field of medicine in the US, but it is believed by many that this was a mistake made by the US Military when they chose it to symbolize their medical division. The rod of Asclepius is more appropriate and can be found on the back of most ambulances, at least that I have seen.
As far as my education and research suggests, the caduceus is a symbol for commerce, thievery (hustlers), balance, and for me, the cue.
To answer your questions regarding my definition of a cue. When I say a cue, I would include any other cylindrical object with a rounded tip used to strike an object in a game, and I would count the maces and strikers used in dead sports such as lawn billiards, trucco, or any other cue sport as a cue.
I would not be surprised to learn that older cultures such as the Indians, Persians, Aboriginals, Greeks or Chinese had cue sports that have died out and been forgotten. Therefore, cues that have been potentially lost to time. The act of striking an object with a stick is a pretty fundamental relationship between man and his environment, as is competition between man and man. As someone who has studied anthropology extensively, it would also not be that surprising to learn that information regarding these sports and their cues has not survived to modernity.
With that being said, thanks for inquiring about my avatar. Please feel free to share knowledge of the collections you come across, as I am always interested in this sort of thing, no matter the age or geographic source.