As defined in Enterprise Information Systems: Contemporary Trends and Issues (Olson and Kesharwani, 2010):
An enterprise information system (EIS) is any kind of information system which improves the functions of an enterprise business processes by integration. This means typically offering high quality of service, dealing with large volumes of data and capable of supporting some large and possibly complex organization or enterprise. An EIS must be able to be used by all parts and all levels of an enterprise.
Because of the combination of “any kind of information system,” with the requirement that the EIS “must be able to be used by all parts and all levels of an enterprise,” means that EIS will always be unique: each is a combination of products and components, some of them large and standard such as SAP, but others highly specific to an industry, vertical, or market. That EIS are unique makes sense also from the perspective that ultimately they exist to support business strategy and value proposition, which are inherently unique.
Some level of internal production is required to create these systems. It’s required in order to set strategic direction, to determine system composition, and usually is required to achieve integration/interoperability between separate components in the overall system. All EIS are internally produced.
For these reasons, I use the term EIS to refer to internally produced enterprise systems built in support of strategic business objectives.