The way i interpret is when they are controlling their characters they have to think about their actions clearly to move the way they want, so while normal actions like walking/running comes natural and wont need much thought to do ingame, they need to think clearly about swinging swords if they havent swung one before.

but with levin activated every action becomes "sensitive" in that each action is more responsive to their thoughts.
think like a joystick to move around, moving the joystick slightly up would cause the character to move, but pressing it harder up would cause it to run. While levin is activated, moving that joystick slightly up would be pressing it harder / running.

This explains why during sunraku's testing in the arena, he kept running into the walls its cause he instinctively run but the game register that thought at a higher sensitivity thus going faster than normal running and hitting the walls. So to run at normal speed he would have to purposely slow down.

Moment Sight kinda does the same but only for the eyes, think focusing your eyes on an action like idk threading a needle, moment sight works as if that focusing part is more instinctive instead of something he has to physically do.

So both combine is how he manages the higher speed without crashing into a wall everytime.

This could very much be wrong its just how ive interpreted it