I think the problem with that video on color was the extreme example used. I'm quite sure not everyone sees the same exact red, but using the extreme example of someone seeing green and just learning to call it red would have other issues. Only way it would work is if that person had all their colors shifted the same, this would require all 3 types of cones in the retina to be shifted the same amount, biologically that would be a hard thing to do. The chemicals in the cones are fairly well characterized, the variations would be in the levels of nerve impulses generated by the cone cells. The other issue would be that if someone had a major shift like that, one end of the spectrum would always be messed up for them in artificial lighting situations, since we design lights to give us the best view in the spectral range the normal eye sees.
If you want a more realistic idea of the way different people might see the same color, go into the custom color picker in any paint type program, pick a color, and then change the R,G,B sliders random small amounts, a few percent change in each slider would be a realistic way to simulate the variations in the 3 cones between people.
On a more advanced program you could do the same thing for a picture, adjust the RGB ratios small amounts to see how different people might see the same picture but with different sensitivities to red, green, and blue.