In the ERG, both are verbs that take non-verbal complements. The be_id_is is an instance of the ‘identity copula’, which takes an NP complement and introduces an EP that relates that NP complement to the subject. We use this both for identity cases (like The author of that book is Kim) and cases that look more predicatives (Kim is an author). (I think there might be even more semantic categories of NP predicates, but don’t know off the top of my head.) The key commonality is that there is no argument role in the NP predicate for the subject to fill, so the verb introduces a predication that links them.
be_c_is, on the other hand, is an ‘ordinary’ copula that supports a non-verbal predicate that does have an argument position for the subject. Syntactically it is a raising verb, identifying its subject’s index with its complement’s XARG (which in turn points to the complement’s subject’s index). From our perspective, it is incorrect to say this copula has a ‘sense’ because it doesn’t mean anything.