http://www.allaboutcircuits.com/vol_1/chpt_15/1.html says:
"... whereas current can only take place where there are free electrons to move."

Current requires electrical charges to move, not neccessarily electrons. Could also be ions, positrons. Quarks also have charge, but do not occur isolated.

 

Except that all of the useful and interesting practical cases involve the movement of electrons in a conductor or a semiconductor. In the case of the semiconductor the movement of electrons is actually mirrored by the movement of holes. You forgot to mention holes.

Positrons, ions, and quarks (isolated or not) don't seem to have too many interesting applications in conductors and semiconductors.

If we're talking Chemistry then you might get some interest in "ion currents", but that seems to be about it unless you want to talk about beam mechanics in a tokamak or the LHC. Interesting no doubt, but hardly within the purview of your average hobbyist.

 

The OP is invited to register to continue this thought.