Consistency in the CAP sense has a precise meaning: all reads see all completed writes. That is, if the database has told some client that their write succeeded

Consistency in the CAP sense has a precise meaning: all reads see all completed writes. That is, if the database has told some client that their write succeeded, then it is not allowed to still serve other clients the old data. Inconsistency in the CAP sense may also cause inconsistency in the sense that you mean, for instance if two transactions are simultaneously accepted, but each transaction violates the precondition of the other. In a consistent database one of the transactions will see the result of the other transaction and fail, whereas a DB without consistency may accept both transactions and end up in a semantically incorrect state.