Umbrella style vs bowens mount octabox

Hi there, I have 2 softbox 32" and 48" that I use mainly for outdoor portraits with an AD200. They are "umbrella style" octabox like the one illustrated below.
While that design can be limiting in orienting the light downwards, it provides an extremely quick arrangement for setup and folding back: you open them like a simple umbrella and just put on the white diffuser sheet. That way your flash is basically pointing inwards and bouce back off the silver interior surface.
The alternative bowens mount would entail to use the bowens mount adapter which would place the flash at the vertex and pointing directly towards the diffuser.

Do you have any direct experience if the fist design arrangement is poorer in terms of quality of the light?

Are you asking if direct (forward facing flash) vs indirect (rear facing flash) is better? Or what the differences are?

If so, it does depend on the design of the unit and how the strobe is positioned in the unit and the material the unit is made of. The picture you show appears to be a very reflective silver surface, which would give a relatively more direct (harsh) reflection. A more textured surface gives a more diffuse (soft) reflection. A white surface gives an even more diffuse (soft) reflection. A diffuser makes the reflection even softer.

The direct configuration is less softened by reflection but frequently there are two diffusers, inner and outer, which diffuses the light twice. Most of the other factors listed above are similar.

Deep designs tend to be more directional. Flatter designs tend to spread the light out more. A larger diameter will usually be softer.

All other things being equal (which they generally are not . See above). the indirect design will tend to be softer.

If you use the bare bulb for the AD200, it doesn't exactly "point" in any one direction. Are you using the AD200 bare bulb or the fresnel head?

I'd be using it in virtually every modifier.

If the diffusion is even across the face of the front diffusion panel, it doesn't make any difference what's going on in the back.

Front-firing softboxes usually require two diffusion panels to get even diffusion; with rear-firing umbrella-style softboxes, the umbrella shape serves the initial light-spreading function.

It doesn't matter how reflective the internal surface is, if the result is still even diffusion across the front surface.

I always use the bare bulb configuration.
So in conclusion the umbrella style arrangement would give softer light (and maybe slighly less power) being all other parameters equal.