How do large deviation events in a stationary process cluster? The answer depends not only on the type of large deviations, but also on the length of memory in the process. Somewhat unexpectedly, it may also depend on the tails of the process. In this paper we work in the context of large deviations for partial sums in moving average processes with short memory and regularly varying tails. We show that the structure of the large deviation cluster in this case markedly differs from the corresponding structure in the case of exponentially light tails, considered in Chakrabarty and Samorodnitsky (2024). This is due to the difference between the ``conspiracy'' vs. the ``catastrophe'' principles underlying the large deviation events in the light tailed case and the heavy tailed case, correspondingly.