Mathematical games are widely used in primary mathematics classrooms internationally, yet there remains limited conceptual clarity about what features define a mathematical game, and whether teachers' understandings align with definitions articulated in the research literature. This study examined how primary teachers across Australia (n=193), Indonesia (n=225), and Italy (n=107) prioritised eight characteristics commonly used in the literature to define mathematical games. Using an anonymous online questionnaire, descriptive analysis revealed strong overall alignment between teachers' judgements and the research-articulated definition. At least 90% of teachers in each country rated enjoyment, clear rules, explicit mathematical objectives, player choice, and interactivity as at least somewhat important. However, teachers weighted characteristics unevenly: enjoyment was consistently the most strongly prioritised feature, functioning less as one criterion among many and more as a threshold condition for whether an activity counts as a game at all. In contrast, features tied to mathematical integrity, particularly the expectation that mathematical knowledge should meaningfully shape gameplay outcomes, were endorsed less consistently, especially among Italian teachers—24% of whom rated this as very important compared with 62% and 61% of Australian and Indonesian teachers respectively. These findings highlight productive tensions between affective engagement and mathematical rigour in how teachers conceptualise games.