Richard, let's have another go at disagreeing with you - politely

You seem to be saying that the different effects are mainly due to differences in physical size, position and distance, i.e. a softbox of the same size at the same distance produces a very similar effect to a beauty dish, and I am saying that it doesn't.
To be honest, there is very little difference in the kind of light that all three deliver, if they are all the same size and at the same distance. The differences are in the control they give, price, convenience etc.
You are also saying that a beauty dish fitted with diffuser effectively turns it into a softbox.
You can get examples of both that are completely different, as you say, but equally you can often get pretty much the same kind of light out of either. The fundamental difference between light sources is their size, and their size relative to the subject (ie position). And does it have a diffuser panel on the front or doesn't it. I think the OP's beauty dish has an optional diffuser attachment, so then it will be essentially a small softbox.
I'm not saying that fitting a diffuser to a beauty dish just turns it into a diffused beauty dish, but that the effect it produces is more like a softbox than a beauty dish - or in other words, although I don't entirely agree with you on that point, I don't entirely disagree either.
Any test that tries to demonstrate similarities/differences will be flawed. I suppose the perfect test would be carried out by a top fashion photographer using a top fashion model, with perfect bone structure, perfect skin and who can repeat a pose exactly, because the slightest change in pose would make the test irrelevant.
That would be difficult and expensive to achieve, so when I carried out the tests on
various light shapers for Lencarta I did the best I could. I used a mannequin simply because she keeps perfectly still, doesn't need tea breaks, doesn't want paying etc., making her the perfect model for comparative tests.
What made her less than perfect is that her skin is plastic, so is her hair and, worse of all, her eyes are painted, so the catchlights you see are not created by the light I used.
This page on the 70cm beauty dish tests includes a shot with a silver finish beauty dish, both with and without the diffuser. If we accept Richard's point that beauty dishes fitted with a diffuser are pretty much the same as a softbox of the same size and shape, then my tests show a useful comparison between the effect of the beauty dish and the effect of a softbox in the same position, of the same size, and at the same distance.
Everything was the same, as even the camera was bolted to a studio stand, and there was no retouching - these shots were resized only. You can even see the beauty dish, with and without the diffuser, in the top right corner of the photos.
I don't think the differences are subtle.