In hydrology, what's the difference between specific and total catchment area?
I will try to bring you some insights. In my opinion, SCA is poorly named.

Actually, the width is not involved in the catchment area which is given by TCA. SCA introduces a diffusive flow factor by considering flow width. It is a one dimensional value so it is not representative of an area and that's why it might be confusing. It gives us an idea of how concentrated are the incoming flow.

Besides this, I think you are also confused by the outlet concept which is often defined in GIS or in some conceptual definitions by a point. But actually, in order to compute a flow at an outlet you need flow velocity and a cross section. If this cross section area is assumed to be rectangular, it can be defined by the width and the height will vary according to the incoming flow. So when the outlet is considered as a point, it is more like a targeter of the outlet that has some dimensional characteristics.

So, for studies at the plot scale or the hillslope scale, the outlet is often defined by a line which is representative of the flow width at the downslope.

For grid based data, a pour point targets a representative area defined by the spatial resolution of your terrain data. If a point has no flow width, this area has one and the flow width is generally assumed to be the cell size as mentionned in your quote.

If you understand that, you may understand that the broader the width, the more diffuse will be the overland flow and therefore the less wet regarding the TWI or the less erosive. Of course, it remains resolution dependent.