6th generation consoles and dithering?

I don't see a newbie/beginner sub-section for the console section, so excuse me if this is not in the right place.

Although I've been following PC/console hardware for many years, I'm still an amateur when it comes to graphics processing.

It's well known that the large majority (all?) of PS1 games use dithering heavily. There are a few informative youtube videos on this subject.

Apparently, it also seems to be used on 6th gen consoles to some extent. The explanation that I've read around the internet is that although their GPUs are capable of processing at a higher color depth, the frame buffer is not large enough to hold the frames at those depths, so the output is reduced to 16-bit + dithering.

1. Is that explanation correct?
2. Does that apply to all these consoles? If not, what's the reason for each one?
3. Does the original xbox also use dithering? There are posts claiming its output is 24-bit.
4. Is it widespread or used in just a few titles?

I became interested in this subject mainly because of the HDMI mods in recent years. The dithering pattern on old consoles is not supposed to be visible when using a composite connection, and specially on a CRT display, but becomes very apparent on modern displays with HDMI.