However, if you're thinking about it that way you're setting yourself up for future confusion. When you say "the IF which roughly contains the earth and the star", you seem to be suggesting that there might be an inertial frame (or non-inertial frame, for that matter) that does not "contain" the earth and the star. That's not right, because everything is always in all frames always - there's no such thing as being in one frame and not another. There is such a thing as being at rest in one frame and not another, but that doesn't change any of the physics. In particular, the basic principle that the time something happened is the time at which it could be observed minus the light travel time between the event and the point of observation works in all inertial frames. Whether the source or the observer is at rest in the frame is irrelevant.