While the concept of serious game has considerably evolved in the last two decades, it still needs to be clearly differentiated from other types of artifacts. Thus, for most outside the domain, there is a degree of confusion about the relationship between serious games and other related applications such as simulators or the re-purposing of entertainment games within educational practices. This article proposes a formal approach towards classifying Games, Toys, Serious Games, Serious Toys, Serious Re-purposing & Modding and Simulators. The aim of this theoretical work is twofold. Firstly, on a practical level, this approach aims at helping actors from different ecosystems, such as health, for instance, to differentiate between these various devices and use them to their best advantage. Secondly, from a research perspective, based on a formal approach, our work aims to contribute to the development of a taxonomy for gamified intervention with serious purposes. This formal approach allows us to demonstrate that unique combinations can be proposed to distinguish each kind of application. In this context, Serious Games can be seen as a specific purpose and not as a synonym for other existing applications.