This paper presents findings from two co-design focus groups conducted with older adults to explore the design of a socially interactive robot that uses large-scale projections (wall or floor) to support active ageing through games, movement, and cognitive engagement. While prior research has examined the use of social robots and exergames for older adults, limited work has investigated how large projection-based interaction influences robot design preferences or how gendered perspectives shape expectations of such systems. We conducted separate co-design sessions with four male and five female older adults, using visual sketches, semi-structured questions, game prototypes, and large projection demos. Thematic analysis revealed both shared and divergent insights. Participants across groups preferred wall-based projections for visibility and safety, and favored a compact, storable robot with voice interaction. However, female participants emphasized aesthetic integration, emotional tone, and privacy, while male participants prioritized challenge, progression, and task control. Our findings highlight how large projection interfaces paired with voice-interactive robots offer a promising medium for learning, training, and wellness in older adults. We discuss design implications for creating inclusive robotic systems that adapt to physical, cognitive, and social needs, and reflect on how such interfaces may generalize beyond exergaming into storytelling, cognitive training, and collaborative learning. These insights contribute to the design of age-inclusive, projection-enhanced HRI systems.