I've heard it's healthy to drink warm water that is close to (internal) body temperature so that energy doesn't go into warming it up and that it doesn't "shock" the system. Does that mean it's unhealthy to drink cold water? I find it helps me cool down on a hot day.
Co-signing to what you said. In the hot summer days I would always try to drink a rather cold water in order to cool down. Just how you said the body warms it up so it basically takes away from the body's temperature. It always works well for me.
I also know from my doctor that very cold water will shock your system and from my dentist that it is bad for the teeth's enamel. That's why I try to not abuse it.