On the basis of the principle of critical-point universality, we examine the peculiar thermodynamics of the liquid-liquid critical point in supercooled water. We show that the liquid-liquid criticality in water represents a special kind of critical behavior in fluids, intermediate between two limiting cases: the lattice gas, commonly used to model liquid-vapor transitions, and the lattice liquid, a weakly compressible liquid with an entropy-driven phase separation. While the ordering field in the lattice gas is associated with the chemical potential and the order parameter with the density, in the lattice liquid the ordering field is the temperature and the order parameter is the entropy. The behavior of supercooled water is much closer to lattice-liquid behavior than to lattice-gas behavior. Using new experimental data recently obtained by Mishima [J. Chem. Phys. 2010, 133, 144503], we have revised the parametric scaled equation of state, previously suggested by Fuentevilla and Anisimov [Phys. Rev. Lett. 2006, 97, 195702], and obtain a consistent description of the thermodynamic anomalies of supercooled water by adjusting linear backgrounds, one critical amplitude, and the critical pressure. We also show how the lattice-liquid description affects the finite-size scaling description of supercooled water in confined media.