Short answer: Cold is pleasant only when your are not already freezing and cold might satiate thirst better because it acts as enhancer of the "water intake flow meter".
Is cold water more tasty than warm water? No, it is actually the reverse as detailed in my footnote.
Cold is pleasant when your body is over-heating and definitely not if you live naked in the North Pole. Over-heating means sweating which means you loose water and therefore feel thirsty faster. Yet drinking cold water will not rehydrate the body more than warm water and drinking water as only a very small impact on the body temperature. So why do we like it?
A study was actually conducted on the subject and answers most of your questions. Here the reference.
The temperature of the body will indeed not change.
cold stimuli applied to the mouth
(internal surface of the body) do not appear to impact on body
temperature and are not reported to cause any reflex shivering
or skin vasoconstriction that influence body temperature.
As you pointed out, the temperature of the ingested water will not affect the overall hydration of the body as cells are rehydrated mostly via the blood stream and the blood temperature will not be affected. Someone could argue that, at identical volumes, cold water (above 4C) contains more molecules (i.e. is denser) than warm water but this difference is likely very slim.
In this paper they also define "thirst"
Thirst is a homeostatic mechanism that regulates blood osmolarity
by initiating water intake when blood osmolarity increases
The problem is that it takes some time before the water reaches the blood stream therefore you need a feedback mechanisms that tells you to stop drinking independently of the blood's osmolarity. This is where cold might play a role.
The cold stimulus to the mouth from ingestion of water may act
as a satiety signal to meter water intake and prevent excessive
ingestion of water
The picture would then be the following
In essence, cold sensation is pleasant in warmth weather, both on the skin and in the mouth and it apparently help in reducing thirst by being some kind of enhancer of the "water intake flow meter".
Reading the comments I just want to clarify some points.
The 5 basic tastes (sweet, salty, bitter, sour and umami) are very distinct from taste sensations (pungency, smoothness, cooling to name a few). The main difference is that taste and "sensations" signals use a completely different path to reach the brain, namely the facial and glossopharyngeal nerves for the former and the trigeminal nerve for the latter.
Is the temperature affecting basic taste perceptions? The answer is yes. How this happens is quite simple if you understand the fundamental concepts of molecular taste perception. Essentially the temperature affects the response of the receptor TRPM5 which is the main player in depolarizing taste receptor cells in the papillae. To put it simply, higher temperatures provoke a greater perception for taste, and this is not only in term of perceived taste but really modifies the amplitude of the response at the molecular level. As an example this is why ice cream does not taste sweet when frozen but only after it melted in the mouth or on the tongue.