I agree with Moab Man. Ring flash might be used for scientific close up work of specimens, but for portraits, no, never.
One definition of the word "modeling" is to show a 3D shape on a 2D surface. Photographers use lighting to create subtle shadows, or gradient shadows, across the face of portraits, to show curves and shape of the subject. A light on the lens axis is a flat light, used for the purpose of hiding all such modeling shadows. A speedlight (unless bounced) is on the camera too, but at least it is about six inches above the lens, and can contribute slight shadows, hardly good, but better. The standard lighting goal is off camera. Off camera is an angle that intentionally produces the shadows. Then we add a weak on-axis light as a fill light, to weaken the shadows (to NOT be be dark or harsh), but NOT to fully remove the intentional shadow shading. The coiled hot shoe extension cords are made to hold a speedlight at arms length, to create some modeling shadows.
It is good to understand the point of lighting portraits. Look at your first link above. Look at the faces. Notice the lighting (that's what photographers do). On the good ones, notice how one side of face is lighted slightly differently than the other (names of the lighting is Short and Broad). This is the entire point, intentional and planned, and is a very good thing.
Ring flashes cause the opposite of that, flat, featureless, uninteresting.
