We use extensive molecular dynamics simulations employing different state-of-the-art force fields to find a common framework for comparing structural orders and density anomalies as obtained from different water models. It is found that the average number of hydrogen bonds correlates well with various order parameters as well as the temperature of maximum densities across the different models, unifying apparently disparate results from different models and emphasizing the importance of hydrogen bonding in determining anomalous properties and the structure of water. A deeper insight into the hydrogen bond network of water reveals that the solvation shell of a water molecule can be defined by considering only those neighbors that are hydrogen-bonded to it. On the basis of this view, the origin of the appearance of a non-tetrahedral peak at a higher temperature in the distribution of tetrahedral order parameters has been explained. It is found that a neighbor that is hydrogen-bonded to the central molecule is tetrahedrally coordinated even at higher temperatures. The non-tetrahedral peak at a higher temperature arises due to the strained orientation of the neighbors that are non-hydrogen-bonded to the central molecule. With the new definition of the salvation shell, liquid water can be viewed as an instantaneously changing random hydrogen-bonded network consisting of differently coordinated hydrogen-bonded molecules with their distinct solvation shells. The variation of the composition of these hydrogen-bonded molecules against temperature accounts for the density anomaly without introducing the concept of large-scale structural polyamorphism in water.