Disposition of fragment shader `gl_PrimitiveID` in unusual circumstances

The fragment shader definition for `gl_PrimitiveID` is as follows:

GLSL 4.6:

> The input variable `gl_PrimitiveID` is filled with the value written to the `gl_PrimitiveID` geometry
shader output, if a geometry shader is present. Otherwise, it is filled with the number of primitives
processed by the shader since the current set of rendering primitives was started.

OpenGL 4.6 is a bit more specific:

> If a geometry shader is active, the built-in variable `gl_PrimitiveID` contains the ID value emitted by the geometry shader for the provoking vertex. If no geometry shader is active, `gl_PrimitiveID` contains the number of primitives processed by the rasterizer since the last drawing command was called. The first primitive generated by a drawing command is numbered zero, and the primitive ID counter is incremented after every individual point, line, or polygon primitive is processed. For polygons drawn in point or line mode, the primitive ID counter i incremented only once, even though multiple points or lines may be drawn.

Let's assume that there is no GS present. The above says "the primitive ID counter is incremented after every individual point, line, or polygon primitive is processed". But "processed" by what exactly? The sentence starts by talking about the "drawing command" being the first primitive. But the previous sentence talked about primitives processed by the rasterizer. Which one is being looked at here? Tessellation generates more primitives than are being input, so it matters.

Also, there is a question of how this behaves with instancing and multi-draw. Does starting a new instance count as a "drawing command" for this purpose? In Vulkan, it explicitly says that it does (starting an instance resets the counter).