Mobile charge carriers (the flow of which we generally view are electricity) are only electrons. There are 'holes' but those are just abstractions of electrons. As far as battery chemistry goes, it's referring to ion's, which are atoms that have more or less electrons than normal, which makes them non-neutral, and so a flow of ions can cause a voltage and thus current. The statement is referring to proton flow, again this is just an ion in this case (I'm not a fan of the term 'proton current'), but the ion is not flowing through the wires, it is only flowing within the battery chemistry. Current is the flow of electrons, and electrons will not flow without a voltage. A voltage is a potential difference between two areas. So, if all positively charged ions (as they lack an electron) move to the positive side of the battery, and all negatively charged ions (as they have an extra electron) move to negative side of the battery, you get a difference where the one side is more negatively charged than the other, and thus you have a voltage. Connect a wire between the two and you have a current.
As far as anyone is concerned, outside of battery chemistry, all electricity is due to electrons. Though, really, even internally in a battery it's still due to electrons if you think about.