Both Belief-MDPs and Bayes-Adaptive MDPs (BAMDPs) are special cases of POMDPs and their state space is augmented with a belief over their unobserved/hidden variables.
In a belief-MDP, the hidden variables can change over the course of an episode. (Eg. Both the position and the uncertainty in the position of the robot can vary during an episode).
In a BAMDP, the hidden variables are usually the attributes of the transition/reward function and are held constant during an episode. (Eg. In a robot locomotion task, ground friction or load - dynamics attributes, current goal location - reward function attributes. Though variables defining these attributes are unknown to the agent and the agent has to infer them, the actual variables remain unchanged during an episode)
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L Zintgraf, K Shiarlis, M Igl, S Schulze, Y Gal, K Hofmann, and S Whiteson. Varibad: A very good method for bayes-adaptive deep rl via meta-learning. arXiv preprint arXiv:1910.08348, 2019.