Two stones can both say "2.00 ct" on the certificate and look like they belong to different weight classes on the hand. That was the thing that got me while researching a 2 carat engagement ring (https://gemroyals.com/collections/2-carat-diamond-engagement-rings). Carat is a weight, not a size, and almost everything else about the stone decides how big it actually looks.
Here's what I wish I'd understood sooner.
Cut matters more than you'd think. At this size, a stone has nowhere to hide. A mediocre cut looks dull or glassy, and a deep cut hides weight in the bottom where nobody sees it. If I were doing it again, I'd rank cut first, then pick color and clarity around it.
The "magic number" is real. Prices tend to jump at round weights like 1.00 and 2.00 because shoppers search for them. A well-cut stone around 1.85 to 1.95 ct usually looks the same face-up as a 2.00 and often costs noticeably less. Ask to see the millimeter measurements, since that's what you'll actually see.
Shape changes the story. Ovals, marquise and elongated cushions tend to cover more finger per carat. Rounds and asschers tend to look a bit smaller for the same weight. Length-to-width ratio matters too, so look at it rather than guessing from a photo.
Color and clarity can be relaxed. In a larger stone, inclusions are easier to notice, so "eye clean" is the goal, not a top clarity grade. Something I noticed repeatedly is that a diamond a grade or two below perfect can look identical in person and free up budget for a better cut. Ask for video, and if possible look at the stone under different lighting.
Natural vs. lab-grown deserves an honest look. Lab-grown stones at this size usually cost far less than natural ones, which can mean a bigger or better-cut stone for the same money. The trade-off is that resale values are generally lower and the market is still shifting. Natural may suit someone who values rarity or a more traditional feel. Neither is wrong, but know which one you're buying and why.
The setting changes everything. A thin band makes a stone look larger. A halo adds visual size but can crowd the center stone, and a hidden halo adds sparkle from the side without the bulk. Finger size and shape matter too. A narrower finger tends to make 2 carats look bigger, and a wider one may favor a slightly longer shape.
A few practical things before you pay:
Insist on a GIA or IGI report and check that the number matches the stone.
Read the return policy before you fall in love with a ring.
Get insurance quotes early, because the cost varies.
Ask about resizing, since some settings are harder to adjust.
On budget, I'd be skeptical of anyone quoting one fixed price. A 2 carat engagement ring can range from a few thousand dollars for lab-grown to tens of thousands for a high-quality natural stone, and where you land depends on cut, color, clarity and setting. Compare current listings rather than trusting any number you read in a forum, including this one.
If you want to browse real examples to compare, Gem Royals (https://gemroyals.com/) has a decent selection of 2 carat. Disclosure: I'm affiliated with them, so weigh that accordingly and compare other jewelers too.
What cut or shape are you leaning toward, and are you open to lab-grown?