"static electricity" or electrostatics is studying charge situations whereby all systems are in ELECTROSTATIC EQUILIBRIUM. That is, they aren't really actively doing any re-distribution.

"Current electricity" is what you shouldn't call it. You should just call it circuits. That is when we focus a study on an application of when electrostatic equilibrium ISN'T the case, and the results are interesting to us.
http://amasci.com/miscon/eleca.html#statcur

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