Robots are promising tools for promoting engagement of autistic children in interventions and thereby increasing the amount of learning opportunities. However, designing deliberate robot behavior aimed at engaging autistic children remains challenging. Our current understanding of what interactions with a robot, or facilitated by a robot, are particularly motivating to autistic children is limited to qualitative reports with small sample sizes. Translating insights from these reports to design is difficult due to the large individual differences among autistic children in their needs, interests, and abilities. To address these issues, we conducted a descriptive study and report on an analysis of how 31 autistic children spontaneously interacted with a humanoid robot and an adult within the context of a robot-assisted intervention, as well as which individual characteristics were associated with the observed interactions. For this analysis, we used video recordings of autistic children engaged in a robot-assisted intervention that were recorded as part of the DE-ENIGMA database. The results showed that the autistic children frequently engaged in exploratory and functional interactions with the robot spontaneously, as well as in interactions with the adult that were elicited by the robot. In particular, we observed autistic children frequently initiating interactions aimed at making the robot do a certain action. Autistic children with stronger language ability, social functioning, and fewer autism spectrum-related symptoms, initiated more functional interactions with the robot and more robot-elicited interactions with the adult. We conclude that the children's individual characteristics, in particular the child's language ability, can be indicative of which types of interaction they are more likely to find interesting. Taking these into account for the design of deliberate robot behavior, coupled with providing more autonomy over the robot's behavior to the autistic children, appears promising for promoting engagement and facilitating more learning opportunities.