Despite agreement on the importance of detecting out-of-distribution (OOD) examples, there is little consensus on the formal definition of OOD examples and how to best detect them. We categorize these examples by whether they exhibit a background shift or a semantic shift, and find that the two major approaches to OOD detection, model calibration and density estimation (language modeling for text), have distinct behavior on these types of OOD data. Across 14 pairs of in-distribution and OOD English natural language understanding datasets, we find that density estimation methods consistently beat calibration methods in background shift settings, while performing worse in semantic shift settings. In addition, we find that both methods generally fail to detect examples from challenge data, highlighting a weak spot for current methods. Since no single method works well across all settings, our results call for an explicit definition of OOD examples when evaluating different detection methods.