It seems that currently, only two types of fusion reactor styles are being used for large scale testing: the Stellarator and the Tokamak.

Well right off, this is not true. The two approaches that are best studied are the "advanced tokamak" like ITER, and the laser-driven ICF approach like NIF. I'd wager that third place goes to the spherical toks, and fourth to the collection of maglif concepts. The stellarator is really not a mainstream approach these days, although the stream of news about X-7 might lead you to believe otherwise.

Shaping such a field seems to be extremely complicated and could apparently only recently be achieved using computational optimization on the coil shape.

That's not the problem. The problem is that when you squeeze a balloon, it tends to pop out through your fingers. We are squeezing the balloon HARD. The computer-aided shaping you refer to only applies to one particular style of the stellarator, the type X-7 uses, it is not needed for other types.

A viable coil arrangement could potentially consist of 6 coils, distributed around a cube shaped container... Why doesn't this work?

Very simply because the magnetic lines aren't closed.

To start with, consider the fact that ions in a plasma at fusion temperatures are travelling at speeds measured as percent of speed of light. If you have an active feedback system that can handle that, I want to buy it!

So in order to keep the plasma inside the reactor when it's going this fast, you want it to naturally stay inside. So in the tokamak, for instance, the magnets are arranged so the magnetic field is running around the inside of the donut. So a particle inside will just circle around and around.

Now with the system you propose, the magnetic field would look like a star. You would have a high field in the center, but some of the magnetic field lines would run through the very center of each ring. And the ions would just follow those lines out of your reactor. Quickly.

So whatever arrangement you make, the key is that the lines are "closed". We've tried open lines in the magnetic mirror and the polywell, and they just don't work. With the mirror they kept adding more and more fields to bend the lines this way and that, but nothing worked, they leaked like a sieve.

It seems that currently, only two types of fusion reactor styles are being used for large scale testing: the Stellarator and the Tokamak.

Well right off, this is not true. The two approaches that are best studied are the "advanced tokamak" like ITER, and the laser-driven ICF approach like NIF. I'd wager that third place goes to the spherical toks, and fourth to the collection of maglif concepts. The stellarator is really not a mainstream approach these days, although the stream of news about X-7 might lead you to believe otherwise.

Shaping such a field seems to be extremely complicated and could apparently only recently be achieved using computational optimization on the coil shape.

That's not the problem. The problem is that when you squeeze a balloon, it tends to pop out through your fingers. We are squeezing the balloon HARD. Its just a hard problem no matter what solution you attempt.

The computer-aided shaping you refer to only applies to one particular style of the stellarator, the type X-7 uses, it is not needed for other types. Tokamaks are very simple to build, pinch devices are even easier.

A viable coil arrangement could potentially consist of 6 coils, distributed around a cube shaped container... Why doesn't this work?

Very simply because the magnetic lines aren't closed.

To start with, consider the fact that ions in a plasma at fusion temperatures are traveling at speeds around 1% of speed of light. If you have an active feedback system that can handle that, I want to buy it!

So in order to keep the plasma inside the reactor when it's going this fast, you want it to naturally stay inside. So in the tokamak, for instance, the magnets are arranged so the magnetic field is running around the inside of the donut. So a particle inside will just circle around and around the inside of the donut. In such an arrangement we call the field "closed", every line of force ends up on itself again at some point.

Now with the system you propose, the magnetic field would look like a star. You would have a high field in the center, but some of the magnetic field lines would run through the very center of each ring. And the ions would just follow those lines out of your reactor. Quickly. We call these arrangements open.

So whatever arrangement you make, the key is that the lines are closed. We've tried open lines in the magnetic mirror and the polywell (the latter is basically what you're proposing, see the field diagrams on that page), and they just don't work. With the mirror they kept adding more and more fields to bend the lines this way and that, but nothing worked, they leaked like a sieve.