This thread appears to be fairly full of loose terms and concepts, which is certainly not unusual given the topic.
I think you just have to decide if you want to actually communicate something, or remain among the ambiguities that are inherent in some of the words we use. Perhaps a person has to decide if someone may legitimately call DNA a “coded medium” –if– it physically operates within its system in the same way as a coded medium physically operates in a system that we would all unambiguously call a code system . Modern humans certainly have enough of them to use as examples.
If such an agreement were being sought, then it would inherently focus attention on the physical characteristics of the system itself, and perhaps many, if not all, of these ambiguities might fall away. For instance (as was already alluded to upthread) a code system first requires a medium of information in order to function. What is required for a medium of information to exist? Does DNA reflect those same requirements within the gene system? Also, an unambiguous code system is (generally a high-capacity) multi-referent system, using spatial-orientation within a single medium to distinguish one referent from another. Does the gene system use spatial orientation within the medium (and its constraints) to distinguish one referent from another? What is physically required for that to occur, and does the gene system reflect those same physical requirements? What about rate-independence? What about the requirement of semantic closure? And so on…
On the other hand, if such an agreement is not amenable to those who argue “DNA is not a code”, then what would be the point of objecting to the word “code” to begin with, and having a discussion about it? It would be a truly meaningless objection and discussion, would it not?