By convention, the source of a river is considered the spring or glacier or other place where water first starts flowing over land that is farthest from where the river empties into a sea or ocean, in the opposite direction from how the river flows out of the mouth.
For example, the Nile runs northwards and empties into the Mediterranean, so its source is considered to be the spring that contributes to the river that is the farthest south (although the guys from Top Gear did an episode where they decided that the Mediterranean was a continuation of the Nile and since it connects to the Atlantic in the west, then the source of the Nile must be the tributary to Lake Victoria that is farthest east, which was just a muddy little spring coming out from under a big rock).
All the other springs and streams and rivers that add to the main river are considered just tributaries and often have their own names, if any name at all.
As for the ultimate source of the water, springs are usually fed by rainfall within certain areas that drain through the ground to a particular point where the water resurfaces. So, the ultimate source is rainfall, except in the case of snowpack or glaciers, where the source is snow (obviously).
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