Ok, I think I see what you want to do now. Keep in mind the tutorial you’re following may be using a different workflow or different features than are on display here. I can’t speak to what you might be seeing there. It may be possible you are watching a tutorial for the Spotlight feature for instance, which performs a little differently.
“Crispness” is going to be a function of your mesh resolution. The more/smaller polygons there are, the finer detail you will be able to capture with the mesh. So if you go to Tool> Geometry> Divide and subdivde your mesh a few times, you should be able to capture much finer detail.
Keep in mind there may be a limit to how much detail Dynamesh can capture–it is generally intended to be used at lower to medium levels of detail. The level of detail you’re trying to add there is generally something that would be added when your need for Dynameshing has diminished.
Your alpha also plays a role. It appears to have a lot of grey in it. Any non-black value is going to cause elevation. Any place in your alpha you do not want to cause elevation should be black.
You may also find that different brushes perform differently with alphas. The Layer brush may provide superior results for what you want to do. Reducing the ZIntensity will reduce how drastically the alpha displaces the mesh. Moving the Focal Shift slider to the left will reduce the radial fade.
Ive given you a lot of information here. If you have further questions after you absorb it, please let us know.
Good Luck! 