What is the definition of "cathode"?
Electric current is an abstract current, the flow of electric charge, not a physical current like, say, electron current, the flow of electrons.

But electric charge is a property of things, not a thing, i.e., electric charge is always "carried" by a thing.  

So, while an electron current is necessarily an electric current (due the negative electric charge carried by the electron), an electric current is not necessarily an electron current.

For example, in a salt solution, we have two species of electrically charged ions present,  the positively charged sodium ion and the negatively charged clorine ion.  Imagine that the sodium ions are moving to the right and the chlorine ions are moving to the left.

Obviously, we have two ion currents in opposite directions but there is just one electric current and it must have a direction.  The direction of electric current is, by convention, the direction of the flow of positive charge.

So, in this case, both ion currents contribute to an electric current to the right.  The first term is due to the positive ions to the right.  The second term is due to the negative ions to the left where the negative sign numerically "flips" the contribution to the electric current.

Think about it this way, if I told you that I was travelling at -60mph west, you'd know that I was actually going 60mph east.  Similarly, a negative charge current leftward is an electric current rightward.

So, the above is all to simply say that the definition isn't lacking.