Some of it will mix with the water along the way, but most of it winds up near the bottom of the glass. With the cold water, the coloring tends to stay towards the top and slowly begins to drift downwards. Even after a minute, both the hot and cold water will still have areas where the color is concentrated and areas where the water is clear. Once we understand think about hot and cold water, this makes good sense. One spoonful of cold water weighs more than one spoonful of hot water.
Cold water is denser than hot water and it tends to sink.
When you put the colored water into the hot water, the coloring sank towards the bottom because it was cooler and denser. As the coloring warmed up, its density was nearer to that of the hot water, letting some of it mix upwards. When you put the colored water into the cold water, it tended to float near the surface because it was warmer and less dense. The colored water floats, just as cooking oil does. Both are less dense than cold water.