Even "get the best cut" isn't universally good advice.
the cut on my diamonds is shit, by most standards. With the naked eye, you can see that the facets are uneven and crooked. That's because they were cut in approximately 1910. I have to rose cut diamonds and a miner's cut. (Basically, that means "old diamonds, cut by hand in ways different than those currently favored). I also have less than half a caret, I believe, which includes those 3 main stones, plus 16 (I just had to count) tiny diamonds set in the filigree. I adore my ring. It cost well less than half a month of my husband's salary. It's unique and spectacular, and as a bonus, since it was an estate piece, I didn't have to worry about my money supporting the nastiness that is the diamond trade and the diamond companies. I wanted something unusual, something second hand, and something with a very low profile (how far it stuck out off my finger) so I wasn't bumping it on things, catching it when I put my hand in my pocket, etc. And my ring is all those things, perfectly. So I don't care that the cut is shit and that the stones are probably not great quality. It's the perfect ring for me and it was a price we both thought was reasonable.
So women want a giant diamond and don't care much about cut, clarity, or quality. Not my thing, but it that's what someone wants, then spending money on making sure it is flawless and perfectly cut is a waste. Some women want a perfect diamond, even it it's a bit smaller. Some want something unique and don't care about the diamonds, and some don't want or even like diamonds.
Of course everyone should set reasonable budget limits, but beyond that, trying to supply some universal law about what is most important is going to fail because it is such an individual thing. Find out what she likes and values most, and then make sure you get a good price for whatever that is.