I know rocksdb is LSM-based and was built by FB to address write amplification on SSDs.
Badger, RocksDB, LMDB, etc. are storage engines. A process uses these storage engines to write data to memory (note: the storage engine may support persistent, volatile, or both types of memory).
A database management system (DBMS) is a higher level concept that often has multiple processes either on a single server or distributed across multiple servers. Simply stated, each process within a DBMS uses the storage engine to read/write to/from memory (persistent or volatile).
It's important to note that storage engines are a specialized area and require different skills from writing a DBMS. It's a big deal when other people write high quality storage engines because it makes it a lot easier to write a DBMS.