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.

Wiki has it right, the cathode is the terminal of a component where the charge flows out. Charge flow (current) is the "standard" definition (i.e. Franklin's one from positive to negative, the opposite of electron flow).

The cathode of a component can change depending on it's state - for instance when a battery is discharging, the cathode is its positive terminal, and when charging its negative terminal (since the charge is now flowing into the positive terminal, rather than out, and out of the negative one, so they are reversed)

Of course, when you switch between talking about electrons and talking about electric current the direction flips.

You can define cathode as "the terminal where the electrons flow into the device", but as we are accustomed in everyday electronics life to talk about current flowing from + to -, the equivalent definition in terms of current is as your Wiki link states: "the terminal where the current flows out of the device".

IMO both definitions are totally unambigous. Their usefulness depends on the context.