Is Phong shading a pixel-first computational model?
I agree that it is a bit superfluous to have the "[i]f a mesh covers more pixels in screen space than it has vertices". Basically, if it doesn't then no interpolation is needed since the color of a pixel can be determined by the vertices it contains. But this is just a degenerate case of interpolation and there isn't much reason to call it out.

Your final paragraph is roughly correct, but it does imply that you need to calculate the Phong reflection model at each pixel using the interpolated normal in the Phong shading case while you don't in the Gouraud shading case. In the latter case, you are linearly interpolating colors calculated using the Phong reflection model at the vertices to produce pixel color values. In the former case, you are linearly interpolating normal vectors calculated at the vertices to produce pixel normals which you then use in the Phong reflection model per pixel.

The Phong reflection model is the "lighting calculation".