Water is not a perfect conductor, it has some resistance. The purer the water is, the more resistance it has. I worked on a radar that used purified water for insulation. There will be a voltage gradient in the water from the voltage source, to the point where the voltage has dropped to zero.
Exactly right, and this is why electric shock drowning is much more of a problem in fresh water than salt. Salt water is a better conductor than the human body and the current dissipates quickly. In fresh water, the human body (a big bag of salty water) is the better conductor and so the current seeks the path of least resistance - you.