Disclaimer: The following definitions are intended for tracing the long road of defining what Database Management Systems (DBMS) are about, therefore the value of the quotes is regarded from the historical perspective.
"The software that allows one or many persons to use and/or modify this data is a database management system (DBMS). A major role of the DBMS is to allow the user to deal with the data in abstract terms, rather than as the computer stores the data. In this sense, the DBMS acts as an interpreter for a (very) high-level language such as APL, ideally allowing the user to specify what must be done, with little or no attention on the user's part to the detailed algorithms or data representation used by the system. However, in the case of a DBMS, there may be even less relationship between the data as seen by the user and as stored in the computer, than between APL arrays and the representation of these arrays in memory." (Jeffrey D Ullman, "Principles of Database Systems", 1980)
"An integrated data management system, that is a true database system, is one in which data can be held nonredundantly while at the same time, a query which requires a mixture of different kinds of data (such as the query that we've been looking at) can be specified in a single command from the highest level programming interface. Any system that does not have this property really should not call itself a 'database' system; 'file management' system would be a more accurate term." (Mark L Gillenson, "Database: Step by Step", 1985)
"The DBMS is a program (or a set of programs) that allows stored data to be integrated, reduces data duplication, ensures data integrity, eliminates program dependency on file formats, and allows even complicated objects to be easily represented and retrieved. In short, a DBMS is the program that 'processes the database'." (David M Kroenke & Kathleen Dolan , Database Processing: Fundamentals, design, implementation” 3rd Ed., 1988)
"A database management system (DBMS) consists of a collection of interrelated data and a set of programs to access that data. The collection of data, usually referred to as the database, contains information about one particular enterprise. The primary goal of a DBMS is to provide an environment that is both convenient and efficient to use in retrieving and storing database information." (Henry F. Korth & Abraham Silberschatz, "Database System Concepts" 2nd Ed., 1991)
"A database management system is a collection of interrelated files and a set of programs that allow users to access and modify these files. A major purpose of a database system is to provide users with an abstract view of the data. That is, the system hides certain details of how the data is stored and maintained. However, in order for the system to be usable, data must be retrieved efficiently. This concern has lead to the design of complex data structures for the representation of data in the database. Since many database systems users are not computer-trained, the complexity is hidden from them through several levels of abstraction in order to simplify their interaction with the system." (Henry F. Korth & Abraham Silberschatz, "Database System Concepts" 2nd Ed., 1991)
"A DBMS is a large collection of computer processes that can be entirely software or firmware that operate in a generalized or specialized computer that provides languages or mechanisms to: Define the logical database component of a specific database either directly or through reference to the IRDS metadata, that is, the data record types, data record elements, and relationships. Define the physical component of a specific database, that is, the storage structure definition, and the techniques for access strategies, data loading and updates, and database backup. Specify interrogations that access data through one or more types of languages such as host languages like COBOL or FORTRAN, and/or natural languages like query-update, report writers, or procedure-oriented. Specify system control facilities, that is, audit trails, backup and recovery, concurrent operations, and the like." (Michael M Gorman, "Database Management Systems: Understanding and Applying Database Technology", 1991)
"Database packages, more correctly called Database Management Systems or DBMSs, such as Oracle and Microsoft Access provide us with all the facilities necessary to design and use an application database. A DBMS can be viewed as having three components: (1) The user interface is the software that talks' with the user in creating and accessing the database. […] (2) The database engine: this is the software that manages the storage and accessing of the physical data stored on disk. (3) The data dictionary, a repository of information about the application database." (Colin Ritchie, "Relational Database Principles" 2nd Ed., 2002)
"A Database Management System (DBMS) is an organised set of facilities for accessing and maintaining one or more databases. A DBMS is a shell which surrounds a database or series of databases and through which all interactions take place with the database." (Paul Beynon-Davies, "Database Systems" 3rd Ed. 2004)
"A software system that enables users to define, create, and maintain the database and also provides controlled access to this database." (Thomas M Connolly & Carolyn E Begg, "Database Solutions: A step-by-step guide to building databases", 2004)
"A product or tool that manages any kind of database, not just relational databases." (Rod Stephens, "Beginning Database Design Solutions", 2009)
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