That's right, protons are just hydrogen nuclei/hydrogen ions. And your right, there is no distinction between electricity and the movement is ionized particles.

The primary difference i see here is that almost always when we talk about electricity we're talking about moving electrons. But there's no reason you couldn't utilize an organism which moves hydrogen ions for some sort of electrical device. I would imagine it's much harder to move protons though as they're immensely more massive than electrons, so I wonder about speed and efficiency, but it could be really cool in some specific applications.

If nothing else, you could siphon off the hydrogen ions as a sort of fuel cell, that outputs hydrogen and electricity.