What is a "light source" in global illumination? In the context of global illumination and the scattering equation, do we need to understand "light sources" as surfaces with a "purely emissive" material on which no scattering is present?

In contrast, any surface in the scene geometry could be equipped with a self-emissive material and hence being somehow a "light source" (it's kind of a philosophical question, I guess).

However, since we often talk about light-carrying paths of "length" $k$, where one endpoint is located on the camera and the other on a light source, I guess that's how we need to see it. Am I correct or is there something I'm missing?
Like @lightxbulb said, a light source (i.e. an emissive surface whose L_e term in the rendering equation is greater than zero) can also have a BRDF.
People usually do not model light sources really accurately down to the Tungsten filament geometry inside of a glass-enclosed gas chamber for incandescent light bulbs. Such light sources usually are modelled as simple geometries, like spheres, capsules, lines, rectangles or more generic polygons, etc.
So, most often, light sources are usually not exactly physically accurate anyways, because it would be too computationally expensive (imagine computing the intersection of a ray with the Tungsten filament in addition to account for a very little bit of volumetric scattering through the gas).
Therefore, in the simplest cases, such an incandescent light source is down to a simple sphere anyways. In that case, it would make most sense to model this light source with a translucent BTDF (Bidirectional Transmittance Distribution Function), because most of the rays will probably not hit the filament but would go straight through the glass.
What we would have to consider, however, is Fresnel and retro-reflection on the glass surface of the light bulb.
But it all depends on how accurately you want to model your light sources.
If, on the other hand, your light source is a milk glass light bulb, then you would probably need a BRDF + BTDF (or more generally a BSDF - bidirectional scattering distribution function) to account for the light that is reflected off the glass surface and the amount of light that is trasmitting through the glass.

In any case, stopping a ray at the light source (as you pointed out) would only be actually physically correct, if the surface of the light source is a black body not reflecting any light but only emitting light due to its temperature.