If electricity is banned on shabbat, how does a person deal with the problem of the body being a conductor? When I hold a person's hands, it creates a circuit. If I let go of their hands then I have just broke the circuit. Isn't this similar to a light switch?
Static electricity and the small amounts of energy created through human contact are not considered a problem.
Tzitz Eliezer 7:10 writes that there is no problem because the static makes no real impact in that small amount of time, there is no precedent for this type of electricity in the times of the mishkan, and this type of electricity cannot start a fire at all.
So this all ultimately ties back to Mavir or (igniting) and how we are forbidden from igniting a flame during Shabbat.
While a person can benefit from fire or electricity which was started before Shabbat, we cannot make or extinguish a flame on Shabbat and we cannot feed a flame on shabbat.
A spark plug igniting the engine of a car being a real world example.
Because the level of power being produced in static touch is at such a low level and the existence of that static is for such a short period of time, it isn't considered a violation as it doesn't leave any detectable impact.
It only becomes Halachically problematic when it can create a real fire which can burn. It all ties back to being forbidden from producing a fire on Shabbat.
Lets compare this to your example of the light switch.
The amount of power being produced in that light switch is above and beyond the underlying levels we experience when we touch hands with a friend or family member. While humans can certainly create circuits which pass a charge between us, we aren't creating a charge which can produce something like fire.
Even something like a battery is producing a greater charge of electricity than any humans do with a handshake.
This is simply from a battery and even that is above and beyond what we produce through touch.
Humans can't produce such things by touching hands or putting on a wool sweater/coat. It's not something we need to worry about on Shabbat.
The body is not a good conductor of electricity, it would be considered an insulator. However, even with a high voltage you can pass electrical current through anything, this is called dielectric breakdown. I doubt this is what was meant by electricity, and certainly doesn't happen when you join hands with someone.
As an aside, electrons are continuously moving through your body, as are other charged particles, like ions which are primarily responsible for neuron function in a body's nervous system.
Halachic electricity would only be an issue regarding man made electricity e.g. battery, mains electricity, etc. And could include several issurim including igniting a fire, 'final hammer blow', molid, etc.
However, if it were even an issue, bodily 'electricity' is certainly something unavoidable and a דבר שאינו מתכוון, and would therefore be allowed.