To create electrical current, you must have charges that can move, a.k.a. “mobile charges.”
Sometimes the charges are comprised of mobile electrons. An example is when there’s current in a wire.
But current doesn’t have to be made up of mobile electrons. When there is current through a battery, for example, the current does not consist of electrons - it consists of ions.
Same goes for electrical current through your body. The current consists of ions, not electrons.
As for the water analogy, it’s O.K. But there are some problems with it:
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Electrons do not “come out of the battery” on one side, and then “back into the battery” on the other side.
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The energy is not due to the kinetic energy of the electrons *moving *in the wires. It is in the fields surrounding the wires.