Given this definition, does this mean that every electric component that has electrodes (i.e. anode or cathode), is a nonmetallic part?
Under that definition, in order to have electrodes a component must have nonmetallic parts. But it can't be entirely non-metallic since the electrodes are part of the component and they are metal. For example, a vacuum tube is generally made from metal, glass, and empty space (vacuum). It has both metallic and non-metallic components.
And what exactly is the use of an electrode? Is it just used to indicate polarity?
The electrodes are typically the path for current to flow in and out of the component. The exact metal used might have an effect on the performance of the part, for example in a Schottky diode.
The name of the electrode depends on the polarity: the electrode where current flows in to the component is called the anode and the electrode where current flows out is called the cathode. In the case of zener diodes, these definitions are somewhat abused.
Are there other types besides anode or cathode?
A device with more than two electrodes (for example a transistor or a triode, tetrode, or pentode vacuum tube) necessarily has electrodes that aren't called the anode or cathode.
In vacuum tubes, there's typically an anode, a cathode, and one or more grid electrodes.
In a transistor there is no anode or cathode, just base, emitter, and collector for BJTs, or gate, drain, and source for MOSFETs.
Can I use electrode as a synonym for pole?
I'm not aware of any case where that would make sense. We normally talk about magnets having poles, and electronic devices having electrodes, terminals, pins, pads, contacts, leads, etc.
if the electrode is dependent on the path for current to flow in and out of the component, then does that mean the cathode could switch terminals of a component if the current is reversed?
In principle, this is true. In practice, we choose one terminal of a device to call the cathode and one to call the anode, based on the "normal" use conditions, and we don't change the names when the current direction changes. In the case of zener diodes, we even name the cathode and anode according to what they would be if the part were a rectifier diode, but we normally use a zener with current flowing in to the terminal we call the "cathode".