Another term that confuses things are projector headlights. They can be lit with either HID or halogen. The cheap, non-HID 'xenon' bulbs should at least be called 'xenon-like' as they are simply tinted halogen bulbs. These can be good quality and attractive, but there's no xenon involved. Xenon illumination is impossible without a regulated power supply.
HID is a generic term, referring to various lamps also seen in streetlights, petrol stations, and sporting venues. Even xenon HID is a bit misleading, the xenon is only used as temporary illumination until the HID component heats up enough to work on it's own. I think all automotive HID lamps are actually some sort of metal-halide lamp with a xenon startup component.
So if you hear any of these terms, and it's important, all I can say is ask more questions.