Something along the lines of "R&D" could be a good name for the demo category. I prefer "category" instead of "tier" because I don't consider one category to be superior over another (except stars, of course), and I would rather have them noted as different groups. The categories need a slight redefinition to reduce subjectivity. Without giving names to the categories, here is how I would define them:
The speedrun category is for TASes that have speed as the main goal rather than entertainment. (Clearly defined any% and 100%)
The entertainment category is for TASes that have entertainment as the main goal rather than speed, and are deemed entertaining. There should not be any mixing between this and speedrun, which is what the current system does.
The star category is for TASes that are the best on the site. (This will remain the same) This could work as a concurrent category so that it is easy to tell what type of starred TAS it is.
The demo category is for runs that are interesting technical achievements (but not trivial or simplistic), but are not deemed entertaining enough to be in that category, and do not fit in the speedrun category. There are a number of rejected TASes that can fit in this category, most notably the gruefood delight.
As for the chess example about early mates, it depends on how it is accomplished. I could make a 2-player version, but it would not be any real accomplishment, nor would beating up a simplistic engine. But if someone could get a notable engine to get an early mate (or at least relatively quick in terms of number of moves), that would be very impressive, but not entertaining because of the long waiting.