I always love it when people don't know what types of sources are. (See what I did there?)
Even when Wikipedia is accurate (which, granted, is usually over 90% of the time), it still cites secondary (or even other tertiary) sources. Very rarely does Wikipedia reference primary sources. Asking the person you're trying to make a point to to wade through multiple layers of interpretations in order to get to what you're trying to say to them is the academic equivalent of saying "No, you!" If you have something to say, say it. If you feel you need to back up what you have to say with a source, cite a primary (or, in rare cases, secondary) source. Don't rely on the person you're trying to prove a point to, to play "follow the link."
In any case, symbols mean what people believe them to mean. If people see a Caduceus and think "medical," then that is what it means. It's like saying a Swastika is representative of peace and harmony. That may be the case, in some places, and some times, but most modern people associate it with Germany's National Socialist Party (Nazis), so that is what it means.
As an ironic aside, the Caduceus was used as a symbol for anyone treating eyesight by any means. We call them Optometrists these days. They're doctors. So the symbol isn't completely off base, even when looking at it from an out of date historical perspective.