I could be wrong here, but the point of colors safe for use on the Internet was such that your images would look the same on a monitor with a lower color depth (for example, 8-bit color), as well as on one with a higher color depth (16 / 24-bit color). What happens because all these colors should (theoretically) be displayed in all color formats.
If you have an image that is not “safe for a website” and you view it in an 8-bit color system, all colors that do not match the display color capabilities should automatically display as the closest color that suits. Basically, you do not need to "convert" it. The bit-depth color does not allow any OTHER colors to be displayed, and so it will have to connect to something that should be the next closest display color.
However, colors that are safe to use on the Internet are times when only 8-bit color is used on displays. I don’t think that you really need to worry now if you don’t know that your audience uses very, very old technologies.
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