Calls for new metrics, technical standards and governance mechanisms to guide the adoption of Artificial Intelligence (AI) in institutions and public administration are now commonplace. Yet, most research and policy efforts aimed at understanding the implications of adopting AI tend to prioritize only a handful of ideas; they do not fully account for all the different perspectives and topics that are potentially relevant. In this position paper, we contend that this omission stems, in part, from what we call the relational problem in socio-technical discourse: fundamental ontological issues have not yet been settled-including semantic ambiguity, a lack of clear relations between concepts and differing standard terminologies. This contributes to the persistence of disparate modes of reasoning to assess institutional AI systems, and the prevalence of conceptual isolation in the fields that study them including ML, human factors, social science and policy. After developing this critique, we offer a way forward by proposing a simple policy and research design tool in the form of a conceptual framework to organize terms across fields-consisting of three horizontal domains for grouping relevant concepts and related methods: Operational, epistemic, and normative. We first situate this framework against the backdrop of recent socio-technical discourse at two premier academic venues, AIES and FAccT, before illustrating how developing suitable metrics, standards, and mechanisms can be aided by operationalizing relevant concepts in each of these domains. Finally, we outline outstanding questions for developing this relational approach to institutional AI research and adoption.