Background. Although the analysis of engagement is crucial for digital entertainment or learning games, the concept of players' or learners' engagement is still confusing. Indeed, in digital games research, several concepts referring to the idea of engagement such as immersion, involvement, presence, and flow are used. Also, while the characterization of engaged-behaviors may be useful for designers or teachers in assessing players' or learners' engagement, the nature and the scope of these behaviors are still unclear. Aim. In this article, based on a multidisciplinary state of the art on the concept of engagement, we define and delineate the concepts related to digital gaming engagement. Results. We characterize engaged-behaviors by identifying four types of engagement: environmental, social, self, and action. We thus refine, disambiguate, and characterize the concepts of engagement and engaged-behaviors. This work therefore constitutes an effective support for analyzing, designing, assessing, and personalizing engaging activities in digital games.