Your diagram definitely supports your observations.
Although not currently using one, and taking your diagram into consideration, having used beauty dishes in the past, this is what I found;
(most of which is in agreement to what you are saying.)
• Unlike a large softbox, the beauty dish is designed to be used to create shadows.
• Depending on how far back the beauty dish is, if the subject moves, the light will change very dramatically as the subject moves out of the centre soft light and into the outer harder directional light.
• Based on the placement (the distance from the subject) of the beauty dish, you are able to contour and etch out the cheek bones and the chin and retain the center softness of the face. Something that the softbox does not achieve. This is happening because of the low contrast indirect light in the centre hitting the subject in the middle of the face, and the harder directional light feathering on the cheeks.
• By lifting it higher than the subjects face, you can create a fuller looking bottom lip whilst retaining a soft middle part of the face but still have beautifully contoured cheek bones.
• By fixing a diffuser sock in front of the dish, it is able to more closely mimic a softbox as some of the hard-directional light is deflected back in before leaving.
• And as you suggest in your question, depending on the size of the beauty dish and the size of the deflector inside, it is possible to bring the dish in close enough to dramatically reduce skin details, but you have to take the inverse square law into consideration where the sudden light fall off will cause very extreme outer edges.
• To really get a broader soft light, you need to move your beauty dish further back and look at the shadow on a wall and place your subject inside this shadow. This will avoid the outer hard light hitting the subject and you will create a more feathered look. Of course, you will have to bump up the light output.
• And finally, the directional outer light is great at bouncing back off a reflector.
So, I don’t believe you are wrong, but it is possible that some of the articles and tutorials on the internet that you have come across may be somewhat confusing or perhaps just misleading because the Beauty Dish can be used for both hard Edgy light, of very soft low contrast image.
And it is definitely not focused light. to get the soft look, you are shooting in the shadow of the light.