Interactive spherical displays offer numerous opportunities for engagement and education in public settings. Prior work established that users' touch-gesture patterns on spherical displays differ from those on flatscreen tabletops, and speculated that these differences stem from dissimilarity in how users conceptualize interactions with these two form factors. We analyzed think-aloud data collected during a gesture elicitation study to understand adults' and children's (ages 7 to 11) conceptual models of interaction with spherical displays and compared them to conceptual models of interaction with tabletop displays from prior work. Our findings confirm that the form factor strongly influenced users' mental models of interaction with the sphere. For example, participants conceptualized that the spherical display would respond to gestures in a similar way as real-world spherical objects like physical globes. Our work contributes new understanding of how users draw upon the perceived affordances of the sphere as well as prior touchscreen experience during their interactions.