Crisp linen is the cloth people imagine from a painting: pale, sharp, a little stern. Stonewashed linen is the cloth people actually keep. Neither one is fake. They are two finishes of the same fiber, and they diverge most in the first month.
Stonewashed
The mill washes the finished sheets until the fiber relaxes. The surface goes matte. Wrinkles look softer. Shrinkage is mostly already done, though a careful listing will still warn you about a small extra relax in your own washer. This is the right default if you have never liked “scratchy” linen, or if the set is a gift and you do not want the recipient to do the breaking-in for you.
Crisp, or loom-state softened only lightly
Untouched linen can feel dry and a bit harsh. It also drapes with more structure and can look extraordinary on a bed you are willing to fuss over. After several gentle washes it catches up to the stonewashed hand. If a listing will not say whether the cloth is washed, assume you are the one who will wash it into softness, and buy a backup set in cotton for the first week if you are sensitive.
A third thing: brushed
Brushed linen is sanded or teased so it feels fuzzy. It is cozy. It is also warmer, and it can pill. Hot sleepers should skip it. If the photo looks like flannel, it is not the airy flax this site is talking about.