What is the name for a computer case that exposes the motherboard when opening?

There isn't a name for those features - hinge door opening downwards with motherboard on the door. But while there isn't a particular name, you can look up particular terms to find such a design.

It didn't exist prior to the case you pictured, the Powermac G3 B&W (Note, the Powermac G3 pre B&W did not have such a case).

From wikipedia on the G3 B&W,

http://en.wikipedia.org/wiki/Power_Macintosh_G3_(Blue_%26_White)

Though still based on the PowerPC G3 architecture, the G3 B&W was a totally new design. The first new Power Mac model after the release of the iMac, it used a novel enclosure with the logic board on the folding "door", which swung down onto the desk for easy access (a design that was also used on all Power Mac G4 models except for the Cube)

So, it was repeated with the Powermac G4,

There have been people modding the case(the modders say it has to be an mATX(Micro ATX) case. And there has even been a PC case based on that design.

So if you look up PC case like Powermac G4, this page comes up, the Nikao Submarine, unfortunately that was in the year 2000-2002. mentioned here and archive.org has page of nikao submarine on the nikao site here It was manufactured in south korea, and was an ATX case.

Nikao doesn't seem to be up anymore. Scrolling down to the bottom of the Nicao site it says "Tradex Information Technology Inc" but they don't seem to be around either.

Googling i've done hasn't picked up anything since, other than people modding or trying to get one.

There is the Dell optiplex GX280 case some have mentioned, that has some merits, but not like that G3 B&W / G4 case. The GX280 opens like a shark mouth minus the teeth i.e. it has to be pryed open(or let the weight of the case pry it open as you push the buttons on the side and lift the 'door' up a bit). Not like a thin door. And the door doesn't open downwards with the case upwards, you have to turn it on its side and the door opens upwards. And the motherboard is not on the door of the case. It's screwed into the base. But it has its merits, it's a good design, relatively speaking it's very convenient to operate on. But you'll have issues modding that case. A google shows the power supply is a very funny/non-standard shape. So, knowing of that case isn't that helpful. It has variations but GX280s I see seem to be small sized too.

if you were to mod a case you'd be better off modding the mac G3 B&W or G4 one, which is what some people have done, and is the case you want. Though even the Mac one is mATX rather than ATX so smaller than ATX, less room on the board for cards. But at least that's still a standard, which is preferable to the Dell.

I see many links of people modding the G4 case, articles dated 2009, even 2012. Though the computer was sold 1999-2004. It is clearly a much loved case design. And the best 'name' for it, for googling it, is the Powermac G4 case. or the model that brought it into being(which you pictured), the Powermac G3 B&W.

Added
Incorporating some comments.
The OP found the Sigma Unicorn case,

It is an ATX. case pictured below, from this hardwaresecrets review. You see it open with motherboard showing on the open door that (as described in this tomshardware review) hinges downwards.