We have a "green detecting" cell in our eyes. This cell does not see any green, therefore our brain knows it's on the "other side" of the spectrum from green. We therefore have purple, a reddish-bluish color which cannot be represented by just one frequency.

You have three types of cells in your eye that see color, called cones because they’re cone-shaped.

Each of the three has a color it’s best at detecting, and we name them based on that. One for red, one for green, one for blue.

But, it’s not that they see one color and that’s it. Any color you could pick will hit all three, and the ratio between the three signals they send to your brain is what turns into a color perception. Also, the color they’re best at isn’t a narrow sliver, it’s a big hump of the spectrum that falls off to either side of the middle color.

Yellow activates a bit of the red cone and a bit of the green cone, but not as much for either as a red color or green color would have.

Same for colors near blue, a nice teal will activate the blue cone and a bit of the green cone.

But for violet, that’s where it gets a bit odd. The red cone has its big hump in red, falling off in sensitivity toward orange and yellow on one side and toward infrared on the other (which none of the cones see). But for whatever reason, way down in violet, the red cone has a little bump in sensitivity again, after not seeing green or blue at all. So violet literally is the blue cone sending a strong signal, but a little bit of the red cone too.

Pigments and computer displays take advantage of this in exactly the same way: add pure red and pure green, and your brain sees it as yellow, because it can’t tell the difference between actual yellow and an exact mix of green and red.

> Each of the three has a color it’s best at detecting, and we name them based on that. One for red, one for green, one for blue.

This is incorrect. The three cones in your eye are most sensitive to violet, green, and yellow-green, and they're not called red, green, or blue cones. The usual name is S, M, and L, for short, medium, and long wavelengths.

> But for whatever reason, way down in violet, the red cone has a little bump in sensitivity again, after not seeing green or blue at all. So violet literally is the blue cone sending a strong signal, but a little bit of the red cone too.

Also incorrect. The L cone does not have any particular sensitivity to violet light.

I’ve seen different data presented, I’m not sure I can explain the discrepancy.

My guess is that you're thinking of some sort of opponent-process model or something like the CIELAB space. But neither of those corresponds to the physical response curves of cones.

And yet if you mix red and blue light together my brain perceives it very similarly to violet, so if the rods and cones are not doing that, what is?

Well one, no, violet is quite a distinct color from magneta (the color you get by mixing blue and red light, i.e., "purple"). They're not hugely far apart, but e.g. on this chromaticity diagram, violet is at the bottom while mixtures of blue and red light form a magenta shade bottom-right of center. Magenta and violet are as different as red and yellow or green and cyan.

But the answer to your question is that you distinguish blues from purples by how different the signals from your M and L cones are. In both colors, the S cones are stimulated. If L > M, you see purples. If L ~ M, you see violet. If M > L, you see blues. At the far violet end of the spectrum, both L and M are near zero. At the far red end of the spectrum, L > M but both are weak, so combining that with blue produces high S, low-but-positive L-M (as opposed to high S, ~zero L-M for violet). The difference L - M is not hugely different between violet and purple, which is why they are similar-ish looking.

You have to understand that 'color' doesn't really exist. Light is just another part of the electromagnetic spectrum like gamma rays, x-rays, microwaves and radio waves. These are all just photons at different wavelengths or energy levels. You brain takes the signal it gets from cells that are sensitive to certain frequencies and creates the hallucination that we call vision. Purple is just what your brain tells you that certain frequency is.

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