"X-ray" is just a word for light that falls within an arbitrary band of wavelength.

Yes, but so are radio waves and gamma rays; it would still sound odd to hear somebody say "radio light" or "gamma light".

I think you have it backwards - light is a term used for specific bands of the EM spectrum. Nobody is calling their FM radio waves light emissions. Unless this is a naming collison with the already established x-rays.

It's an established term in physics. E.g., https://lightsources.org/

Everybody says "ultraviolet light" even though it's invisible.

Not all light is visible.

Right, hence "visible light". So I don't see the problem with saying x-ray light either.