Deuteranopia confusion lines
Is this plane indeed random?


Answer : I don't think so.   

HUMAN CONE RESPONSE

Let's take a look back at human cone response curve.
This is cone response curve for normal people :




  But picture for deuteranopia is quite different. Confusion lines are intersecting at one point. But this is not a point nearby green color, it is opposite. Why?


In case for Deuteranopes, let's take a look at their cone response :



As you can see Deuteranopes have their green cone sensitivity shifted very close towards red.
This also why Protanopia and Deuteranopia is what we know as Red-Green color blindness, where they have difficulty in distinguishing shades of red from shades of green since its cones sensitivity is very close to each other (almost similar). 

So for deuteranopia, why the intersecting point is it not green ? Well, despite the actual missing cone is green, its sensitivity is shifted towards red. Also, this "intersecting point" is called copunctal point.

CONFUSION LINE :

Now about confusion line, indeed each cones in our eyes is sensitive to each RGB primary (even though it's a little bit misleading). And yes, RGB color space is in 3-dimensional. But confusion line is mapped based on chromaticty diagram from CIE 1931, not RGB. Some specific transformation and normalization method need to be performed to convert 3-dimensional RGB into 2-dimensional chromaticity diagram.

As we know before, Protans and Deutans have their cone sensitivity close to each other, thus their copunctal point (point where the fundamental color is missing, point where all the lines converge) is also close to each other.

Here's the copunctal points (1) for all dichromats :

COPUNCTAL POINT     x         y
Protan             0.747     0.253
Deutan             1.080    -0.800
Tritan             0.171     0.000


CIE is quite tricky, they can have negative value as well (RGB can't have negative value). This copunctal point is very important since it's the fundamental information of the missing wavelength from colorblind, and this missing wavelength is converted into CIE colorspace and then mapped into the diagram. So, even though the deuteranopes confusion lines looks a bit confusing, it is the correct mapping from actual missing point in wavelength.

(1) Wyszecki &amp; Stiles, Color Science (2nd ed.), 1982, Table 1 (5.14.2) p. 464