Actually, both are quite possible. In the general case, for an arbitrary elliptical orbit, what you'll tend to find is that B is true (granted there will be some precession, but not usually in line with the Sun). However, it is possible to set up an orbit (such as a Sun-synchronous one) in which A is true. But orbits such as this require planning and precise positioning.
To understand why A does not have to be true (and in fact, why A is not the general case), you should bear in mind that when orbiting a body, your orientation is irrelevant. The Sun ensures both Earth and the satellite will orbit around it, but it has very little influence over the specific way they are pointing.
Imagine that Earth did not rotate on its axis; you'd see the same stars in the same positions every night. However, you'd still experience the Sun rise and set (once per year). The sun doesn't drag the orientation of orbiters (at least, not in a significantly mention-able way). All that is important from the satellite's perspective is Earth, which has the dominant sphere of influence.
As I mentioned earlier, however, it is possible to establish orbits that follow the Sun. A sun-synchronous orbit can be very useful as it allows a satellite to always be in view of the Sun and thus continuously collect solar power. This is possible due to a combination of several effects including the small amount of sway the Sun's gravity has over the satellite. However, to establish this type of orbit requires the use of a specific angle of incidence (which varies based on altitude). In Low Earth Orbit, the inclination is around $98^\circ$.
An important thing to note is that while most orbits do not generally follow the Sun, they technically do precess (meaning they usually do not look exactly like B but it's a good approximation/generalization). This precession happens at different rates depending mostly on altitude, inclination, and eccentricity of the orbit. But that's just extra info in case you were curious.
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