And then you have the organizations that confused the Staff of Aesculapius with the Caduceus carried by Hermes, apparently because they both involve a rod and snake(s), so close enough?
This includes organizations like the U.S. Army Medical Corps and countless physician organizations, all proudly displaying a caduceus in their logos and symbology.
Unfortunately, unlike the Staff of Aesculapius, the caduceus never had anything whatsoever to do with medicine or healing. It was instead the symbol for trade and commerce, so maybe a bit of a Freudian slip there by the physicians.
The official explanation for the caduceus is that it represents the speed with which patients were transported to hospitals.
In my opinion, it was more likely an artistic choice. If you shrink a Staff of Aesculapius down to the size of a collar pin, it becomes nearly invisible, and very hard to identify. You could take two of them crossed en saltire, but that might be mistaken for the crossed weapons of the combat services.
Also, the guy who designed the emblems was extremely fond of bilateral symmetry. Which the caduceus has, and the staff of Aesculapius does not.
From everything I’ve read, it was pretty clearly a mistake due to confusing the two symbols, and all of the various subsequent justifications were post-hoc rationalizations after the erroneous symbol had become so well-established.
As for a collar lapel pin, you could simply put it on a background like the EMT logo above (with crossed bandages).
