If you're on a train moving at a constant speed, and you start bouncing a basketball up and down, then to you, it is traveling a path that is perpendicular to the floor. To someone outside the train, the basketball travels a diagonal path. Light is no different in this respect. So the conclusion is that the angle that the bouncing object follows is relative to the rest frame of the observer. Special Relativity is no different from pre-relativistic physics in this regard. I understand the intuition: In the light clock, when the clock is moving, it appears that the light must be "aimed" ahead of the mirror. But the same thing is true of the person bouncing a basketball inside a moving train: From the perspective of someone at rest looking at the train move past, it seems that the person with the basketball must "aim" it slightly ahead of the point on the floor where he wants it to hit. But the person aboard the train isn't doing anything differently than if the train were at rest.