Using comoving coordinates gets rid of the expansion of the coordinate distance between two objects that are each at rest relative to the Hubble flow. It doesn't get rid of the expansion in general, for objects that are moving relative to the Hubble flow. Light rays are moving.
The middle diagram has as its axes the comoving coordinate $\chi$ and the clock time $t$ of an observer at rest relative to the Hubble flow. Suppose that someone in galaxy A sends a radio signal to cosmologically distant galaxy B, then B immediately replies and sends a signal back. A receives the reply after some round-trip time $\Delta t$. Now say they do it again, so we have $\Delta t_1$, $\Delta t_2$, and so on. These times have to get longer and longer. But if the light cones looked like the standard 45-degree lines on this $\chi$-$t$ graph, then the $\Delta t$'s would all be the same.
Usually when we talk about shapes in GR, we mean intrinsic shapes. A geodesic such as the world-line of a light ray is intrinsically straight. (It gets a little more complicated when you talk about the shape of the entire light cone, since it's possible in some cases to rule a curved surface with straight lines.)
