The interesting point is DB/DC does contain both transaction manager and database manager. So this region supports to CICS to access IMS database. Except BATCH it supports for all kinds of transactions.
Note: Pure BATCH runs under MVS address space. So BATCH jobs cannot access IMS database, unless they used as BMP (Batch).
DBCTL- Since this region does installed only database manager. It supports for CICS programs and BMP batch programs. Also for DB2 stored procedures through ODBC.
DCCTL – It does support for BMP,MPP, and IFP programs. Since this region does installed only transaction manager.
Please note that BATCH, BMP(BATCH) and BMP(txn) all are different.