We investigate accuracy and freshness of status updates from a large number of energy-harvesting devices that monitor two-state Markov processes and access the medium using the slotted ALOHA protocol without feedback. Using a Markovian framework, we analyze the average value of a generic state-dependent penalty function that grows whenever there is a state estimation error. The age of incorrect information (AoII) is an example of such penalty function. We propose an accurate and easy-to-compute approximation for the average penalty. Numerical results demonstrate the benefits of optimizing the transmission probabilities according to the process state transitions and current battery levels to minimize the average penalty. Minimizing a state-independent penalty function can be highly suboptimal in terms of average penalty when one of the process states is critical, i.e., entails a high penalty if wrongly estimated. Furthermore, minimizing the average penalty does not guarantee a low probability of misdetecting a critical state period.