Existing methods for dealing with knowledge updates differ greatly depending on the underlying knowledge representation formalism. When Classical Logic is used, update operators are typically based on manipulating the knowledge base on the model-theoretic level. On the opposite side of the spectrum stand the semantics for updating Answer-Set Programs where most approaches need to rely on rule syntax. Yet, a unifying perspective that could embrace all these approaches is of great importance as it enables a deeper understanding of all involved methods and principles and creates room for their cross-fertilisation, ripening and further development. This paper bridges these seemingly irreconcilable approaches to updates. It introduces a novel monotonic characterisation of rules, dubbed RE-models, and shows it to be a more suitable semantic foundation for rule updates than SE-models. A generic framework for defining semantic rule update operators is then proposed. It is based on the idea of viewing a program as the set of sets of RE-models of its rules; updates are performed by introducing additional interpretations to the sets of RE-models of rules in the original program. It is shown that particular instances of the framework are closely related to both belief update principles and traditional approaches to rule updates and enjoy a range of plausible syntactic as well as semantic properties.