Which observer is correct in his definition of time?
Neither? Both? or both Neither and both?
If the relative velocity between the two frames is constant (i.e., inertial), then the effects are reciprocal and BOTH correctly see the other's clock ticking out time more slowly than their own. However, if a force is involved and one of the frames is accelerating (i.e., changing its rate of speed OR direction), then the effects become real. It takes energy to move ahead in time. Velocity and therefore acceleration are 2-component vectors containing both speed AND direction. Change either the magnitude of the speed or the direction and the frame accelerates.