Yep. I didn't look at your links, but I spent most of a day looking at the whole RL history of this for something else (not related to Star Trek) earlier this [year]. The fact that RL history went the way it did stinks, but it is not in anyway Cryptic's fault that this symbol should be in use in the game. it already was what it was long before Cryptic and Star Trek crossed paths.
A valid point by the OP. (I learned something today)
I think the symbol only really came into use on TNG. Poking around the interweb, McCoy always just used a generic sciences insignia on his chest. The 09 movie appears to have gone with a medical cross to denote medical.
So, here's a challenge for ThomastheCrypticCat. How about a new 2409 medical logo using the correct rod?
Also, languages, as well as symbolism, symbols, representations, and meanings of concepts, words,and ideas, evolve with time. The Caduceus has come to represent the medical field, and that is what people understand when they see it, therefore that is the meaning of it. Historical meaning and current meaning are not necessarily the same thing. You can get bogged down by the evolution of thought, by not keeping up with it, or you can adapt and evolve with it. It doesn't matter to me which one you choose to do, but understand that is your choice, not the choice of the rest of the world.