Confusion Database System and DBMS
Often the two terms are used interchangeably, so you may find Microsoft SQL Server or Postgres being referred to as either a DBMS or a database system.

Where a distinction is maintained "database system" usually refers to the overall application or set of applications that constitute a project. So for instance in the case of a business management suite that has a few applications which uses a few on-premises SQL Server databases for online storage, and has a data warehouse in some cloud database for reporting, and utilities for maintaining the warehouse and running reports; the application suite is a "database system" that itself uses one or more instances of one or more "database management systems" as part of its construction.

A "database system" in that respect might use several different database management systems in its design. A single application suite may use OLTP style stores for the main data, OLAP optimised stores for reporting, and NoSQL stores for "soft" supplementary data (and/or perhaps caching).