My notion is if at the same distance, the reflected umbrella always wins, in a few ways (size, soft, spill). However, the shoot-through umbrella can be, and normally is used much closer, when it could win.
Here is a comparison of shoot-through and reflected umbrellas (same umbrella, at same distance). If both are metered at the same distance from the fabric, the reflected is one stop brighter than the shoot-through, which means 2/3 of the light is reflected, and 1/3 goes through. The black cover prevents that 1/3 from being any problem for reflected, but nothing prevents the 2/3 from being a problem for shoot-through (spill reflecting all over the room). However, shoot-though is normally only used very close to the subject, like 12 or 24 inches (closer than reflected is able to do is its entire purpose), so then, inverse square law says a 6 or 10 foot path to and from spill wall reflections will be way down, relative to a 1 or 2 foot forward distance.
Some say that a shoot-through is softer light than reflected. I have tried really really hard to verify this, and I cannot. I believe it to be clearly the wrong answer. I always get the opposite result, and it seems obvious. Shoot-through suffers from the edge of the umbrella being wrapped back, that side portion not being aimed at any subject, and therefore the effective diameter is obviously smaller, quite a bit smaller relative to the light hitting the subject. Just look at them - Reflected is simply about twice larger (effectively), because the outer rim redirects light back to the subject, instead of the rim pointing away from the subject. Simply just look at the example shape comparison at right, and this is quite clear. Smaller is definitely not softer.
Yes, shoot-through obviously can be placed closer than reflected, and then closer certainly can be larger and softer, which is only reason why we might be interested in it. It can be a lot closer, which is why it is done. But it is not softer when at the same distance, which would allow any comparison of the umbrella modes.
Try this.. Get a couple of eggs from the refrigerator. Handpick a couple with rougher texture to their shells. Put them in a saucer. Fill the frame with that dish, and the shadow under it. Three feet is about as close I can get the reflected umbrella, so set it up carefully measured at 3 feet to the fabric (at fabric flat apex, at the shaft). Also take another picture, same picture, but carefully measured to be three feet from the shoot-through fabric. I agree both at 3 feet is not actually equal, but they are simply not equal, how else could it be compared?
Not a huge difference, but a noticeable difference. First picture (reflected) is obviously the more soft light. Has to be softer, it is obviously larger at the same distance. Brighter too, and tremendously less spill. To me, shoot-through is a pretty special case, and I consider the reflected umbrella the better choice in every case where it can be used.
A test situation for below: Metered at 5 feet carefully measured from umbrella fabric at apex, using makeshift plumb-bob over a tape on the floor. ISO 200. Full power SB-24, zoomed 24 mm. Umbrella at full shaft length (less important to shoot-through, the edges are not going to contribute anyway).
B&H Impact 32" umbrella.
Reflected f/8 + 8/10 stop
Shoot-through f/5.6 + 8/10 (1.0 stop less)
Smith Victor 45" umbrella
Reflected f/8 + 9/10 stop
Shoot-through f/8 + 1/10 (0.8 stop less)
Photogenic Eclipse 60" umbrella
Reflected f/8 + 9/10 stop
Shoot-through f/8 + 2/10 (0.7 stop less)
Note for the larger umbrellas, the fabric is the same measured distance, which means the stand is slightly closer.
The worst part is all that rear spill from shoot-through, the majority of the light is scattered out the back. If the light meters one stop brighter reflected, that one stop is double power, meaning 2/3 is reflected, and 1/3 goes though. So for shoot-through, most of its power spills out the back side, all over the room (instead of aimed at subject) - but frankly, which is no particular issue when used very close as intended (inverse square law), closer than reflected could be used. If very close to the subject, that direct light is immensely brighter, and the reflections from any distant wall dimension are relatively insignificant. But if not close, spill bounce back varies with the room and its dimensions of course. The convertible black cover on reflected umbrellas does not aid the reflected light, but it simply prevents the spill out its back side.
The reflected umbrella is stronger, larger, softer, and much better behaved about spill. I would choose reflected mode in every opportunity when distance allows reflected to be used.
But if the umbrella must be very close, within 12 or 18 inches, then shoot-through is of course the only convenient choice. And all that rear spill must travel a far distance then, relative to the close forward distance (to the walls and back), so it will be weak then, not an important factor (when very close).
Sure it can. Look at the numbers above. A reflected umbrella is a reflector, like bouncing from the ceiling or wall. We don't measure the ceiling or wall to see if it is too large for our speedlight. The larger umbrellas have corresponding longer shafts (my shafts are: 32" is 22 inches. 45" is 27.5 inches. 60" is 32 inches). So the angle to fill their area remains about equal - they look about the same from the flash. Larger is a few inches farther (to it and back), but the larger umbrella can cover the same area (like a group, or a full length standing portrait) when it is placed a little closer (to be softer), so it all works out. Different fabrics might vary slightly in reflectivity, but size is no difference in respect to power. Distance is what counts.
However, a too-large umbrella can be a serious pain to work around in a small space. 45 inches is a very nice size.
With speedlights in reflected umbrellas, I use maximum shaft length to try to fill the umbrella,, and I use 24 mm zoom, which Nikon flash manuals say the field is 78x60 degrees spread. The umbrella at maximum shaft will appear slightly wider, about 90+ degrees wide, but customary engineering practice is that these field widths are at 50% intensity points. And of course, there is no sharp edge on the fall off - the spill is substantially wider than that. If you zoom longer, or use less of the umbrella shaft, the focused speedlight won't fill as much of the umbrella. The output will be about as bright, but a little less soft, and less wide.
For a shoot-through umbrella up close, the flash might as well be a little closer on the shaft, since filling those outer umbrella edges is not going to affect your desktop scene anyway.
I don't worry much about this, 24 mm seems right and works fine on a speedlight, and I've never been aware of any problem. If concerned with side spill, you could take a flash picture of the edges of the shadow of the black cover on the wall behind the reflected umbrella (to see it). Anything outside that shadow is spill. The appearance of that result will vary greatly with exposure level, so judge it at your regular portrait exposure. No matter what you do, you will see some side spill. If a reflected umbrella, side spill is aimed away from the subject, but possibly towards the camera lens, so use a lens hood, and pay some attention the spill light is not into the lens.