Is the direction of a laser flash affected by the source's movement direction?

If the spaceship is moving with a constant speed with respect to another inertial observer not in the ship, then occupants of the ship would see the laser light moving straight up from the ship, as in your first scenario. From the point of view of the observer outside the ship, the light would travel at an angle with respect to the direction of motion. Check out the diagram in this derivation of time dilation in special relativity: How special relativity works

A basic postulate is that the laws of physics are the same in all inertial reference frames. So the occupant of the ship cannot do any experiment to prove he is moving at any particular speed. Thus, if he shines the laser straight up, it has to move straight up from the ship, otherwise he would be able to prove he is in uniform motion.