There is not an any error. This is just the physics/calculus of the situation.
In the first case (zero phase) the input acceleration is sin(wt). So starting from an initial velocity of zero and initial position of zero (the initial states of the integrators), the body would accelerate up to some maximum positive velocity, then decelerated down to zero velocity. So the velocity is always positive and in one period the mass has net displacement, x. Then you do it for another cycle and it is displaced another net positive x.
With the acceleration phase shifted by pi/2, the acceleration is first positive then negative then positive and then negative. So the velocity increases to some positive value then decreases to a negative value back to a positive values and then back to zero. The resulting change in position is zero, it first increases, gets to some maximum and then returns to the starting point.
