There's been a lot of confusion in regards to Xenon, and its understandable, even I get confused. Let's see if I got this right, I've been corrected before (and rightfully so, I don't want to spread incorrect information), so here we go.
HIDs and Xenons are not the same, they're both high intensity bulbs that usually require a ballast, HIDs contain <insert name> gas, while Xenons contain Xenon gas. The differences is that HIDs don't last as long, and run fairly hot. Xenons last longer, takes time (usually around 10 secs but can vary) to fire up, but cost more.
Bi-Xenons refer to the headlight lens projector, where it doubles as a high beam and low beam, and in some countries (including Canada), they are used as DRLs as well.
Here's how they look like projected onto a wall, pics grabbed from Google Images, not mines.
OEM projectors (usually for Halogen, like the VT, Mazda3 (some trims), usually lower end models have these or halogen reflectors):
****As you can see, they project in a fairly linear but blurry direction.
HIDs/Xenon in halogen projectors:
****Usually they don't project as well due to the design of the projector which was made for halogen bulbs. Some projectors have better cutoffs but the cutoffs aren't always parallel.
****These have sensors that actually autofocus and may adjust if the steering wheel is moved, also auto-aims. Straight cutoff, usually lower on the left side, this is so the light won't be projected onto incoming traffic, while its raised to the right, giving you more projected light to where it matters.