Differential Privacy (DP) is a mathematical framework that is increasingly deployed to mitigate privacy risks associated with machine learning and statistical analyses. Despite the growing adoption of DP, its technical privacy parameters do not lend themselves to an intelligible description of the real-world privacy risks associated with that deployment: the guarantee that most naturally follows from the DP definition is protection against membership inference by an adversary who knows all but one data record and has unlimited auxiliary knowledge. In many settings, this adversary is far too strong to inform how to set real-world privacy parameters. One approach for contextualizing privacy parameters is via defining and measuring the success of technical attacks, but doing so requires a systematic categorization of the relevant attack space. In this work, we offer a detailed taxonomy of attacks, showing the various dimensions of attacks and highlighting that many real-world settings have been understudied. Our taxonomy provides a roadmap for analyzing real-world deployments and developing theoretical bounds for more informative privacy attacks. We operationalize our taxonomy by using it to analyze a real-world case study, the Israeli Ministry of Health's recent release of a birth dataset using DP, showing how the taxonomy enables fine-grained threat modeling and provides insight towards making informed privacy parameter choices. Finally, we leverage the taxonomy towards defining a more realistic attack than previously considered in the literature, namely a distributional reconstruction attack: we generalize Balle et al.'s notion of reconstruction robustness to a less-informed adversary with distributional uncertainty, and extend the worst-case guarantees of DP to this average-case setting.