Different Coloring Procedures - Ink soaks into the surface that it colors, but paints do not soak in -- they cover the surface with a coating. In fact, a primer is usually applied to the surface before painting so that the paint won't soak in. For this reason, paint sometimes chips off the surface that it covers, but this never happens in the case of ink.
Surface Protection - Paint not only colors a surface, but also provides the surface with a protective covering that will keep it from deteriorating when assailed by such agents as wind and rain. Ink offers no such protection.
http://www.ehow.com/facts_7225030_difference-between-ink-pai...
This is another one of those "where do you draw the line" questions where the line is rather broad and ill-defined. It's a continuum. When it comes right down to it, the intended purpose and traditional usage in the field determine what is an ink, what is a paint and what is a stain.
What I don't understand is I'm looking at my CISS tank next to my printer, and the ink is an almost non-opaque liquid and it is not thick like paint at all.
My guess is that the ink in this video is not used for commercial computer printers, but rather larger machines that are used to print news papers and the like.
The ink there is probably custom made, you send the factory the material that you want to print on, and the moustached dude (the guy that is very fast with two spatula) will figure how to get the colours the designer wants on the material, and then they make the ink needed to mass printing on that material.
I've made a arcade machine once: http://tinyurl.com/cnv4h88 it took very hard work to get the colours right on the cabinet (and they actually, are still not right! they are just "acceptable" and I had ran out of money to do more experiments).
Each material reacts very differently to ink and paint (the arcade cabinet is partly painted, partly printed), and the results vary a lot without testing.
Note on the video how he says that after some testing, they add wax for example, this is a important part to make the ink reach some sort of desired effect on the material, maybe the material is think and will be crumpled (like candy wrapping or a t-shirt), then you need a very thin ink, that will not break, or a thick but flexible ink, that will stretch out as needed before breaking. But you might want to print on a very hard material (for example a plastic marquee) then you might prefer a ink that sticks very well on the material (try printing on metal...) and that is resilient (what happens if someone throw a rock on your road sign?)
I always has been a coder, but I would never figure when I went to design school (I am bachelor in design now) that I would find it fun :)