Mixed-initiative co-creative storytelling games have existed for some time as a way to merge storytelling with play. However, modern mixed-initiative co-creative storytelling games predominantly prioritize story creation over gameplay mechanics, which might not resonate with all players. As such, there is untapped potential for creating mixed-initiative games with more complex mechanics in which players can engage with both co-creation and gameplay goals. To explore the potential of more prominent gameplay in mixed-initiative co-creative storytelling games, we created Snake Story, a variation of the classic Snake game featuring a human-AI co-writing element. To explore how players interact with the mixed-initiative game, we conducted a qualitative playtest with 11 participants. Analysis of both think-aloud and interview data revealed that players' strategies and experiences were affected by their perception of Snake Story as either a collaborative tool, a traditional game, or a combination of both. Based on these findings, we present design considerations for future development in mixed-initiative co-creative gaming.