If you're talking about an object in orbit compared to an object at rest on the Earth's surface, both gravitational time dilation and the other type you describe come into play, because the objects are in motion relative to each other as well as being at different altitudes in the Earth's gravitational field. Many other comments in this thread have addressed this.
> In this case, it is the accelerated frame of reference in which time "slows down" relative to the non-accelerated frame.
Not really. The key difference is not acceleration; it's the fact that the super space ship is in motion relative to the center of mass of the galaxy, while the Earth is not. (Strictly speaking, the Earth is too, but its motion with respect to the galaxy's center of mass is so slow that it can be ignored in this scenario.) Similarly, if I sit at rest on the Earth's equator and you move westward around the equator at the same speed as the Earth is rotating (about 450 meters per second), then when we meet up again, my clock will have less elapsed time than yours, because you will have been at rest with respect to the Earth's center of mass, not me (because I am rotating with the Earth, but you are not).
In the first example, yes, it is the combination both of gravitation and acceleration based time dilation.
In the spaceship and the earth example, I can't see what your reference of the galactic center of mass has to do with anything. If you take the space and earth out of the galaxy into empty space, the result would be very much the same. You could say there would be some slight differences because of the extremely minor differences caused by the gravitation effects of the galaxy and the acceleration due to galactic orbit, but being that relativistic speeds are require to see the big differences(we were talking about a spaceship capable of high fractions of C here), I can't see what you were getting at. It is the accelerated frame in which time is "slower" relative to the rest frame, it has nothing to do with gravity.
> I can't see what your reference of the galactic center of mass has to do with anything
The galactic center of mass defines a reference frame that is special with respect to this problem, because that frame is the one that makes manifest the time translation symmetry of the spacetime. However, I do see that I left out an important piece of that: see below.
> If you take the space and earth out of the galaxy into empty space, the result would be very much the same
Yes, because empty space has a similar time translation symmetry, as long as the Earth is at rest in it. If the galaxy is included, the Earth has to be at rest relative to the galactic center of mass; I see now that I was implicitly assuming that it was, without saying so (I did hint at it when I commented about the Earth also rotating around the galactic center, but too slowly to make a difference). So you're right that the galaxy itself isn't really relevant; but the underlying time translation symmetry is.
The point is that what makes the Earth observer in these scenarios have the longest proper time is the fact that he is the one who is "at rest" with respect to the underlying time translation symmetry of the spacetime. Acceleration only comes into it because that is the particular mechanism you chose to make the space ship move relative to that time translation symmetry. In flat spacetime (which is essentially the idealization you're adopting here), accelerating is the only way to move relative to the underlying time translation symmetry. But this does not generalize: there are plenty of examples in curved spacetime where unaccelerated observers can be the ones with shorter elapsed proper time, because they are moving with respect to an underlying time translation symmetry.
If you're talking about an object in orbit compared to an object at rest on the Earth's surface, both gravitational time dilation and the other type you describe come into play, because the objects are in motion relative to each other as well as being at different altitudes in the Earth's gravitational field. Many other comments in this thread have addressed this.
> In this case, it is the accelerated frame of reference in which time "slows down" relative to the non-accelerated frame.
Not really. The key difference is not acceleration; it's the fact that the super space ship is in motion relative to the center of mass of the galaxy, while the Earth is not. (Strictly speaking, the Earth is too, but its motion with respect to the galaxy's center of mass is so slow that it can be ignored in this scenario.) Similarly, if I sit at rest on the Earth's equator and you move westward around the equator at the same speed as the Earth is rotating (about 450 meters per second), then when we meet up again, my clock will have less elapsed time than yours, because you will have been at rest with respect to the Earth's center of mass, not me (because I am rotating with the Earth, but you are not).