As mentioned in the article, referring to "electricity" is kinf of vague. Ionized hydrogen floating about is a plasma; a collection of particle with a positive net charge. Therefore if some H+ ions move in a particular direction, that will be a flow of charge: an electric current.
Similarly H+ ions in aqueous solution can carry a current, as in electrolysis.
No, it doesn't have a positive net charge. In a plasma, the electrons are still part of the substance, but they are no longer bound to the nuclei, and are free to move about. Similar to current in an electrolyte, current in a plasma is the result of electrons moving one direction, while the nucleii are moving in the opposite direction.
Similarly H+ ions in aqueous solution can carry a current, as in electrolysis.