It looks like the S, M, and L cones peak at blue, chartreuse and orange, respectively. If so, how do we see colors past 575 nm and before 445 nm? If the cones really respond to purple, green, and red, (as I previously thought) why do they peak at colors closer to green, as shown in the image?
The answer is actually present in your chart: color perception depends on the ratios of activations of different photoreceptors. For example, as you go "more red" from 575, the responses of the Green photoreceptors are falling off more quickly than the Red ones. The same is true for any colors in between, and on the other end, where 400nm light would activate only the blue receptors, whereas at 445nm there would be some mix of activation of the green.
From the context of Fisher information it is actually interesting that photoreceptors don't actually provide much color discriminability at their peaks.
For example, around 575 nm, the "red" photoreceptors actually don't provide very good color discrimination: the response to 565 or 585 is going to be almost identical, for example. The way you can perceive a difference, though, is that the response from the green photoreceptors will be very different at those two wavelengths (just consult your chart!). In vision, as in all the other senses, the information is really held in the slopes!
The cones responsive to long wave lengths (L cones) are simplistically called 'red cones', those sensitive to intermediate wave lengths (M-cones) are often called 'green cones', and those sensitive to short wavelengths (S cones) are often referred to as 'blue cones'. Red, green and blue cones are hence terms reflecting simply their peak absorbance. However, as your plot shows, there is a great amount of overlap in the absorption spectra of the red, green and blue pigments in the cones in the retina.
To your question: How can we perceive see light with wavelengths larger than 575 nm, and shorter than 445 nm
This is because the cones are sensitive to a whole range of wavelengths outside of their characteristic peak absorbance in the visual spectrum. However, the sensitivity does drop beyond the peak, so red light with wavelengths >445 nm will be perceived less and less clearly, up until the point it becomes undetectable (infrared). Likewise, shorter wavelengths than 445 will loose intensity and eventually become invisible (ultraviolet).
The green and red opsins are basically the same proteins with only a slightly shifted absorption spectrum - indeed, these opsins split late in evolution due to a tandem repeat and small point mutations.
Color vision is nonetheless very specific and millions of colors can be discerned by someone with normal trichromatic vision.
This is accomplished by color opponency:
The opponent color system has 2 channels, namely a red versus green and a blue versus yellow system. The input are the photoreceptors in the retina: red, green and blue cones. A third achromatic channel is used to code visual brightness (Fig.1.). The neurophysiological machinery is located in the retina and higher visual structures such as the lateral geniculate nucleus.
This model is referred to as Hering's opponent color theory (Gouras, 2012). It states that the blue channel suppresses yellow (and vice versa) and red suppresses green (and vice versa). The advantage of this system sharpens color contrasts between opponent colors (Hurvich and Jameson, 1957). In other words, the broad flanks of the absorption spectra of the opsins are sharpened at their edges by opponent colors. This mechanism causes the fact that we cannot see yellowish blues or reddish greens etc. Instead we can only see mixed colors between channels, e.g., yellowish greens or reddish blues (purples).