Coherence vs Consistenty
You say: “As far as I know, “doxastic coherence” means consistency in beliefs and that is all that is of relevance here.” It is not. Doxastic coherence is coherence between beliefs considered as opinions; its contrast set is “epistemic coherence” i.e., coherence between beliefs as knowledge claims. Further, in some senses, while there can be no coherence if there is no (logical) consistency between sentences within a particular theory, ‘coherence’ does not mean ‘consistency of beliefs’. One could have a coherent set of beliefs even if it comprises of mutually exclusive or even (logically) inconsistent subsets of beliefs. That is because (logical) ‘consistency’ is defined within a particular meta-logic, whereas ‘coherence’ applies to discourses. They do not, therefore, mean the same. One can have a coherent discourse, where multiple logics are used in different fragments of the same discourse. While some logics could allow material implication (or some other logical connective), other logics used within the same discourse might consider material implication an invalid rule of inference. This inconsistency (one and the same rule of inference is both valid and invalid) in two different fragments of a discourse does not make the discourse incoherent. In such a context, ‘doxastic coherence’ names a coherent discourse but does not pick out a (logically) consistent set of opinions.