Previous interpretations of language models (LMs) miss important distinctions in how these models process factual information. For example, given the query "Astrid Lindgren was born in" with the corresponding completion "Sweden", no difference is made between whether the prediction was based on having the exact knowledge of the birthplace of the Swedish author or assuming that a person with a Swedish-sounding name was born in Sweden. In this paper, we investigate four different prediction scenarios for which the LM can be expected to show distinct behaviors. These scenarios correspond to different levels of model reliability and types of information being processed - some being less desirable for factual predictions. To facilitate precise interpretations of LMs for fact completion, we propose a model-specific recipe called PrISM for constructing datasets with examples of each scenario based on a set of diagnostic criteria. We apply a popular interpretability method, causal tracing (CT), to the four prediction scenarios and find that while CT produces different results for each scenario, aggregations over a set of mixed examples may only represent the results from the scenario with the strongest measured signal. In summary, we contribute tools for a more granular study of fact completion in language models and analyses that provide a more nuanced understanding of how LMs process fact-related queries.