Rings statistic has been widely used to investigate the network topology in numerically simulated network-forming materials in order to rationalize their physical and mechanical properties. However, different topologies arise depending on how rings are counted, leading to incomplete or even contrasting physical interpretations. Solving this critical ambiguity is of primary importance for the correct assessment of material properties. Here, we show how such differences emerge in water, a complex network-forming material endowed with polyamorphism and a directional network of hydrogen bonds whose topology is correlated with the anomalous behavior of water. We probe the network in the liquid state at several thermodynamic points under equilibrium conditions, as well as during the out-of-equilibrium first-order-like low density to high density amorphous transformation. We study three schemes for counting rings and show that each of them provides complementary insightful information about the network, suggesting that a single counting scheme may not be sufficient to properly describe network topologies and to assess material properties. Our results provide a molecular description of the rings in supercooled water and of the amorphous-to-amorphous transformation kinetics, hence shedding light on the complex nature of water. Nonetheless, our results expose how delicate the proper choice of method for counting rings is, an issue with important consequences for rationalizing the properties of network-forming materials at large.