It's perfectly possible to do physics without using the words "frame" or "frame of reference" at all. The modern view of GR is based not on the idea of "frames", but rather on the idea of manifolds. Take your favorite modern text (Caroll and Wald come to mind), and count the number of times the various phrases "frames" and "manifolds" are used. Bettter yet , try to actually look at the significance of the terms to the presentation of the theory. While arguing about the meaning of what constitutes a frame of reference may be fun for people who like to argue, it's not terribly productive in my opinion. There is to my mind a dearth of definitive definitions in textbooks of precisely what a "frame of reference" is. As evidence of the lack of definition, I point ot the fact that we've had this argument before, and nobody has actually quoted an exact definition. I believe this lack of definition is due precisely to the fact that the underlying concept is no longer central to the theory. Rather than get stuck in rehashing "frames", it's better to realize they aren't terribly important anymore, and proceed onwards and learn something about manifolds.