Signal is the emission of interest, however it occurs. Noise is the other stuff. In common use though people mentally (and literally) drop the first word from "modulated signal" and seize upon signal. It's normal and writers should make an effort to use unmodulated or, probably better, natural in such descriptions. Emission isn't a bad word for this context, but it's still all encompassing and easy to mentally go the wrong direction when reading it.

Is not the signal in question modulated by the motion of the source, as most signals in astronomy are?

Probably, but I think the post you are responding to is talking about the sort of modulation that artificially alters a radio carrier wave in order to encode information.

The post I am responding to suggest calling that signal unmodulated, and I think that would be wrong.

It probably depends on the domain. For telescope people, a “signal” is anything other than noise and systematic error in your measurements.

I’d speculate that this terminology came from telephony or radio communications - even the term “noise” is a giveaway. But it has migrated into astronomical language in a slightly different way.

I’ve seen many stretch the term over the years, but many in the scientific community know it’s watered down enough to mean “something was emitted”