The Redshift in General Relativity

In one picture, there is no velocity. It's just that space itself is expanding. In the second picture, where there are two frames and spaces expanding between them, then the photon that leaves one galaxy is stretched out by the expansion of the universe. And what general relativity tells us is that there's no unambiguous way to decide between these two pictures. That's why relative velocity for very distant objects is not well-defined in general relativity.