Language models are widely used for different Natural Language Processing tasks while suffering from a lack of personalization. Personalization can be achieved by, e.g., fine-tuning the model on training data that is created by the user (e.g., social media posts). Previous work shows that the acquisition of such data can be challenging. Instead of adapting the model's parameters, we thus suggest selecting a model that matches the user's mental model of different thematic concepts in language. In this paper, we attempt to capture such individual language understanding of users. In this process, two challenges have to be considered. First, we need to counteract disengagement since the task of communicating one's language understanding typically encompasses repetitive and time-consuming actions. Second, we need to enable users to externalize their mental models in different contexts, considering that language use changes depending on the environment. In this paper, we integrate methods of gamification into a visual analytics (VA) workflow to engage users in sharing their knowledge within various contexts. In particular, we contribute the design of a gameful VA playground called Concept Universe. During the four-phased game, the users build personalized concept descriptions by explaining given concept names through representative keywords. Based on their performance, the system reacts with constant visual, verbal, and auditory feedback. We evaluate the system in a user study with six participants, showing that users are engaged and provide more specific input when facing a virtual opponent. We use the generated concepts to make personalized language model suggestions.