Rule-based ML approaches include learning classifier systems, association rule learning, artificial immune systems, and any other method that relies on a set of rules, each covering contextual knowledge.
While the rule-based machine learning is a type of rule-based system in which it is often hand-crafted and other rule-based decision makers. This is because the rule-based applies some form of machine learning algorithm to automatically identify some useful rules, rather than a human being needing to apply prior domain knowledge to manually construct rules and curate a rule set.
Yes, rule-based learning is reinforcement learning in the established use of the term. You may run into some opposition from those doing active research today, as the "hot" portions deal with deep learning applications.
Your application has a well-defined game tree to search; you can direct the reinforcements with a mathematical structure that corresponds directly to the game. This is a machine learning application, along with well-established learning algorithms.