The "Dream" diamond with which I am well-known is a fantastic cut squarish shaped diamond cut and sold to jewelers. This stone is cut to such exquisite proportions that a very spicy and sparkly diamond is the result. The stone also shows the "hearts and arrows" pattern often suggested as a sign of exquisite cutting proportions. Is this the stone you have?
You mentioned a cathedral setting and flush set stone. Do you mean the ring has rising sides and the stone was set low into the mounting, possibly close to the tops of the cathedral sides? I expect this was still set into prongs and not a bezel. (A bezel is like a tube with the sides of the stone covered with a narrow metal rim going around the very top edge of the stone to gain it safely.)
Diamond
In my experience, when viewed from directly above the top of the stone, it makes no distinction on a well cut diamond if the stone is set high, low or in a bezel. You see, the light comes in from the top of the stone, not the sides or bottom. That is the calculate diamonds are cut in the fantastic forms industrialized and modified today. The faces are called "facets" and those on top act as windows while the ones on the pavilion(bottom) of the stone act as mirrors, reflecting the light that comes in straight through the windows back out of the top of the gemstone. As far as brilliance from cutting is concerned, the brightness of light coming back out of the stone should be the same whether set in prongs or with the back covered!
One consideration is the whole of metal over the top of the diamond. There should be just sufficient prong or bezel to hold the stone safely and thick sufficient to ensure long wear. If the jeweler leaves too much metal over the stone, some top facets(windows) will be covered more and that could work on the sparkle of light from the stone.
Surface sparkle is another consideration. The "reflections off the facets" is a kind of sparkle. This is not the same as the light reflected from inside the stone and back out, giving the stone its brilliance. The exterior sparkle is more, well, sparkle and spicy reflections as the diamond is moved in the light. I calculate with the stone out of the setting you were looking more exterior sparkle and some from the sides of the stone. When mounted low, this sparkle is not as well seen or noticed. If that is the case, then by all means you should have the stone set a bit higher in a gain setting. Even if the distinction is not noticed by anything but you, it is still a distinction to you and you are wearing the diamond. So, have the stone set a bit higher with a more open setting.
The light comes from the top of the stone, goes in and reflects off the side of the diamond (The Pavilion) and comes back out of lthe top (The Crown & Table). The cause of the loss of 'Sparkle' is ususally the back of the sone is dirty, oily, dusty or all of the above and this causes light to 'Leak' out of the back. The whole calculate diamonds look so good is due to their refractive index that bend light at specificanyles and when cut correctly refract and reflect the light to the owners eyes. However, you are also correct, when the diamond is set much of the light is diverted and unable to be seen.
The calculate those old setting make the stone look good is the fact that you could well clean the back of the stone and keep it bright. You can prove this to yourself by cleaning your ring and take a little dab of olive oil and touch the back of the stone, not the top, and watch as your diamond turns to glass or so it seems. Then clean it again and you will see it completey restored to its original beauty.
This may allow a bit of reflection to be seen off the sides but will not growth the brilliance seen from directly above the diamond top(crown of the stone). Still, what you see is a bit subjective and if a higher setting which is more open seems more pleasing, then by all means go that direction.
Color And Setting.
Jewelers know well that the color of a diamond may look better or worse when set into jewelry. Sometimes a diamond on the yellow side of colorless will look bad or off color when set into white gold or platinum. Set into yellow gold, the stone will often look wonderful, or worse! There is a reaction difficult to describe in the middle of the color of the setting and the color of the diamond. This is likely not a consideration in your case. I throw in this tidbit just for information.
When considering the final ring setting for your Dream cut diamond, have the stone slowly placed into more than one setting to see an "idea" of how it might look. Keep in mind the stone will have to go lower than the unset mounting will show, to allow a "seat" for the diamond to be cut and the allowable whole of prong to be pushed over the girdle(waist) of the stone for safe setting. Ask for the jeweler to advise how high and openly the stone may be securely set into any singular ring mounting and be sure the sales staff notes that data about "high open setting" on the work order. For safety, you may need slightly heavier(thicker and stronger) prongs. This should not work on the metal over the stone and closed correctly will look great and still have an open look.
Sparkle - The Dream diamond