There is theory that light cone shape does not depend on the reference frame in which it is viewed. So why we draw light cones near black hole differently?
In general relativity, frames of reference are local, not global. Each of the light cones in your diagram corresponds to a certain local frame of reference. An observer using that frame of reference would draw his/her own light cone as the undistorted one, and would draw the other ones as distorted.
In GR, an observer in one local region of space has no unambiguous way of defining the speed of a distant object. Therefore there is no unambiguous way to say whether the speed of light is wrong, if the ray of light we're talking about is far away from us.