Bayesian optimization under contextual uncertainty (BOCU) is a family of BO problems in which the learner makes a decision prior to observing the context and must manage the risks involved. Distributionally robust BO (DRBO) is a subset of BOCU that affords robustness against context distribution shift, and includes the optimization of expected values and worst-case values as special cases. By considering the first derivatives of the DRBO objective, we generalize DRBO to one that includes several other uncertainty objectives studied in the BOCU literature such as worst-case sensitivity (and thus notions of risk such as variance, range, and conditional value-at-risk) and mean-risk tradeoffs. We develop a general Thompson sampling algorithm that is able to optimize any objective within the BOCU framework, analyze its theoretical properties, and compare it to suitable baselines across different experimental settings and uncertainty objectives.