This paper studied the finite-time stability (FTS) and global Mittag-Leffler stability (GMLS) of a class of fractional inertial Cohen-Grossberg neural networks (FICGNNs) with S-type distributed delays. A unified modeling framework based on Lebesgue-Stieltjes integrals was adopted to encompass both discrete and continuously distributed delays. By establishing transformation between the fractional-order integrals of the state function with and without S-type distributed delays, new criteria for the corresponding FTS and GMLS were rigorously derived under appropriate assumptions. The theoretical results were further verified through numerical simulations.