5) Cut (part 2): Diamonds are rated on the QUALITY of their cut. The higher the quality, the more that sparkly, or "on fire" they are. The terms used vary by jeweler, but generally the highest quality is deemed "Ideal Cut." There is a huge tradeoff between carat and this cut. You can get a huge rock that looks like a piece of granite, or a small piece that looks radioactive, probably for the same price.
My personal opinion is that getting better cut and a smaller diamond is the way to go. Quote from my wife's co-worker during a meeting: "I'm sorry, but your ring keeps distracting me because it's so sparkly".
6) Cut (part 3): If you really want to maximize your savings you have to understand how jewelers rate cut quality. Symmetry of the cut, polish, girdle thickness go into it, but also (and most crucially) the dimensions of the diamond. There is an optimal mathematical relationship between the width, height, and table that maximizes the reflection of light (and thus creates that "on fire" look). Here's where it gets really interesting. Essentially a diamond looks like this:

Focus on these 2 numbers:
a) Table: Width of the table (typically described as a % of Max Width)
b) Depth: Total height of the diamond (typically described as a % of Max Width)
Consider this. These %s describe the mathematical relationships between them that defines an "ideal" cut, with ranges for each. But the ranges ALSO define volume, which defines carat, which drives cost!
So, think about it; if you have a diamond that has those numbers within the ideal cut defined ranges, but the:
a) the face is on the small end of the range for ideal (52.4% - 57.5%)
b) the depth is on the small end of the range for ideal (59%-63%)
The result is a diamond that is ideal cut, but that is wider and flatter than other ideal diamonds of the same carat count. But here's the thing. Your woman is not going to look at the table or depth of a diamon they only look at the width! By optimizing this, your woman gets a bigger looking ring and you spend less money.
Last point. There's also price variations by dealer. I spent ~40 hours looking at rings online, comparing the prices at different vendors, and the cheapest one I found was
UnionDiamond.com - about 3-4% cheaper than the next best source I could find, including the diamond district options here in New York. I don't have any relationship with them, other than the fact I bought my ring there and saved myself some cash, without any customer service issues.
So that's how a management consultant buys a diamond ring.