Generative Artificial Intelligence systems have been developed for image, code, story, and game generation with the goal of facilitating human creativity. Recent work on neural generative systems has emphasized one particular means of interacting with AI systems: the user provides a specification, usually in the form of prompts, and the AI system generates the content. However, there are other configurations of human and AI coordination, such as co-creativity (CC) in which both human and AI systems can contribute to content creation, and mixed-initiative (MI) in which both human and AI systems can initiate content changes. In this paper, we define a hypothetical human-AI configuration design space consisting of different means for humans and AI systems to communicate creative intent to each other. We conduct a human participant study with 185 participants to understand how users want to interact with differently configured MI-CC systems. We find out that MI-CC systems with more extensive coverage of the design space are rated higher or on par on a variety of creative and goal-completion metrics, demonstrating that wider coverage of the design space can improve user experience and achievement when using the system; Preference varies greatly between expertise groups, suggesting the development of adaptive, personalized MI-CC systems; Participants identified new design space dimensions including scrutability -- the ability to poke and prod at models -- and explainability.