Borgatti and Everett's model (2000) remains the prevailing standard for identifying categorical core-periphery structures in empirical data. However, two issues have troubled this model since its inception. Firstly, there is the issue of formalizing the ideal core as a complete block or clique, which can be excessively stringent in practice. The second concern relates to the existing approach for capturing inter-categorical tie densities, which consistently penalizes solutions despite having perfectly matched densities. Building upon advancements in the direct blockmodeling literature, this paper extends Borgatti and Everett's model to address the identified shortages. To tackle the issue of inter-categorical densities, we replace the approach of correlating each cell value with the sought density with novel ideal blocks for exact and minimum densities. To address the constraint of a fully connected core, we introduce the p-core, a proportional adaptation of the k-core/k-plex solution, offering greater flexibility in defining the level of cohesion required for core membership. Through the application of these novel extensions to classic network examples, we showcase their utility in evaluating the core-peripheriness of empirical networks.