[tomnipotent]

You're conflating DBMS with RDBMS.

I interpreted the OP as DBMS=database, which absolutely includes application code that stores and retrieves data in proprietary formats.

Even a linked list mmap'd to disk can be a database, just maybe not a very good one.

[eternalban]

A RDBMS is simply a Relational Database Management System. A quick tour of CS history reveals ancient curiosities such as Hierarchical Database Management Systems. And there is more:

https://www.studytonight.com/dbms/database-model.php

[tomnipotent]

There are many kinds of databases. Graph, key/value, relational, hierarchal etc. That doesn't change the fact that any app that writes code to store and retrieve data is creating their own database or using someone else's.

[eternalban]

There are many kinds of data models. A DBMS is a DBMS, and typically a specific DBMS supports a specific data model. A log structured file is at most a data store.

To be fair this is a somewhat fuzzy categorization (DBMS, DB, Data Store) and it can cause confusion. An DBMS is a system, just like it says right on the acronym tin. A DBMS is a system that provides capabilities such DDL, DML, auxiliary processes, etc. to manage a database. There are various data models, e.g. a triple store, for databases, but the data model X is a orthogonal to XDBMS.