The main goal of replacing whey with warm water is to slow down acid production. The starter culture bacteria convert lactose into lactic acid. This lactic acid is absorbed by the casein in the curd, which in turn leeches out colloidal calcium phosphate. Put simply, this results in less elasticity or a crumblier texture. When you remove whey and replace it with water, you're removing acid producing bacteria, but more importantly, you're removing lactose. Consequently, there's less lactose for the starter bacteria to ferment, resulting in a higher drain pH and higher initial pH (later on, out of the press). Thus the cheese will be more elastic, less crumbly. Hot water is used because it helps to raise the temperature as well as slow acid production. The water is added right after cutting and an initial stir.
When adding cold water, you do it at the end of scalding and cooking, right before draining. The main goal is to make use of warm curd's ability to absorb water that's at a lower temperature. The lower the temperature you dilute the whey to, the higher the final moisture content of the cheese will be. At the same time, lactose moves in the opposite direction, resulting in less lactose in the curd. So even though that won't have much affect on drain pH (most important for elastic texture) it will still slow down future acid production.
