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Not the most precise answer but my understanding of this. If someone with more experience in this matter would answer it would be great!

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The perceptual color looks "whiter" (i.e. is more yellow when the first curve says it should still be strongly red, e.g. see the top at 1371 degrees C which is about 1700 K) to human eyes, and cameras - but the latter may render it still differently(!) e.g. cameras will typically saturate before the eye does and render metal surfaces that still look brightly yellowish to the human eye as white - because the color-selective receptors in either case (cone cells for biological eyes, CCD pixel elements for cameras) are not perfect notch filters. There is overlap in their ranges of responsivity, so light that stimulates one "cone" is also slightly stimulating the other two, "whitening" the perceived color, and this becomes more pronounced as the light gets brighter, which it rapidly does for a black body source of escalating temperature.

These colors were obtained from a 0.40 wt. % carbon, alloy steel, as seen through a furnace peep hole during average daylight conditions.

I am not an expert about this particular question but black bodies are called as such because they absorb any incoming light without reflecting or scattering it back. So metals, which have colours and reflect light quite well will probably change the perceived colour slightly. In addition, if you have some additives in the metal (which is often the case, like carbon in iron to make steel), I suppose these atoms will add some glow of their own, like different flame colours for different burning materials.

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But if you take the 1371 degrees C hot metal and put a prism or spectrometer, you will see that the result matches the Planckian curve at least approximately. Heck, if you want to see the true spectral color, don't pay attention to the bright surface, rather, pay attention to the incident light cast on nearby objects which will be less intense due to absorption and scattering: you will see that it looks effectively like a slightly-orange strong red, as consistent with the Planckian color locus with that color temperature.

The two temperature graphs refer to different things. The first is the true spectral color of the light emitted by the hot object. The second is the perceptual color of the surface of the hot object.

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For 1250 K most of the radiation is in longer wavelengths than the visible. At 1750 K the red visible wavelengths are present , hence "red hot metal". As temperatures go higher, more short wavelengths enter.

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So it appears that the colors relate to temperature, but are they exactly the same? Do heated metals have those colors because of black body radiation, or is it for some other reason?

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The scale in your second figure shows the continuous accessing of smaller wavelengths as the temperature goes up. From red to yellow visible wavelengths are increasingly accessed. At the temperatures listed on the right, the amount of energy in the visible spectrum is very small, but enough so that it is recorded in the plot. It says in your link:

A black body doesn't exist and is theoretical, so are the colors shown on the first graphic. Every chemical element has its own emission spectra. Moreover, anything too bright for our eyes look whiter that it is actually. That's why we say that metal looks white around 2500°F even if the emitted light is orange.

Most materials, including metals, obey the black body curves whose areas contain the total energy radiated and are a function of temperature.

Many metallurgical furnaces are very good if not perfect black bodies , It has nothing to do with color or composition. As described above , the observation opening does not increase or decrease the temperature color in the black body / furnace. Everything ( walls, hearth, work , etc) is the same color ( at equilibrium.). This is the principle of the optical pyrometer ; You focus the instrument on anything in the furnace and match the color to a reference spot in the viewer. The pyrometer is used to determine if work is at temperature or if there are any temperature variations in the furnace ( for heat-treat furnaces). Melting, steel making furnaces are a little different because they are always heating or have cold fluxes or alloy additions most of the time. So they have different colors/temperatures. And as far as color 1900 F is as close to white hot as the human eye can measure; you can only tolerate to look in for a few seconds , if you about 10 feet from the opening. That is also about how long it take for clothing to start smoking. I means the heat /color chart shown above is way off.

As the whole visible spectrum is accessible to the eye in this case, they seem "white" by the color perception mechanism of our eyes.