So I suggested brilliant earth in your last post, bought my wife's engagement ring from them, and work with lab grown diamonds so let me give you a little primer.
Lab grown diamonds are often called synthetic, but if it's not another stone, like sapphire, CZ, or moissanite, it's real, actual diamond. Don't let anyone tell you otherwise. Lab grown diamonds are cheaper than mined ones but they are not as cheap as other stones grown in labs. There's probably a perfect middle ground in the lab grown realm between his pricing and your style concerns that is just what you both want.
They are cut and graded just like any other diamond. Some lab grown diamonds are so perfect and devoid of flaws that is how gem appraisers can tell that they are man made. These are still expensive, but you can still get a much better diamond for your money this way. Each stone should come with a quality certificate with all the gritty on its flaws.
No diamond is free of impurities, there will always be nitrogen or other gases in the crystalline lattice no matter what and this is what gives diamond faint hints of color sometimes. Green diamonds were bombarded with radiation at some point. Fun fact: chocolate or brown diamonds are just shitty diamonds that the gem market found themselves having to sell so they marketed their way into making them a luxury. There is a very very small market for incredibly pure diamonds for use in electronics and quantum computing research.
Diamonds are grown in two ways, HP/HT and CVD. HPHT mimics the natural formation of diamond by subjecting a carbon source to temperature and pressures like in the earth. CVD works by bombarding a carbon source (methane in our case) with microwaves and other gases to etch away impurities. We work in the industrial sphere so I'm not sure what either way does for quality in jewelry. I don't know how much of the synthetic diamond market is done via CVD in that sphere but you can do all sorts of stuff via CVD, like dope diamond with boron or nitrogen to change its color and properties.
I like diamonds over sapphires and rubies because they are the hardest material on earth. This means that they will withstand the wear and abuse of daily life better than any other stone. Sapphires come close, but after a lifetime of scratches, bumps, wearing it while gardening, etc a diamond will polish up and be just as gorgeous as it was when he put it on you for the first time.
Just don't go beating it with a hammer.