Special relativity deals with perceived time from the observers perspective, which is why it fails to explain clock slowing with GPS. When a clock is zooming around on a satellite the clock speed isn't just apparently different, it really is different. When that clock comes back down to rest at Earth the clock is no longer synchronized with that which was at rest on Earth.

Lorentz's time dilation calls for that type of effect, while special relativity does not (despite conjuring up a Lorentz equation in its paper, the meaning of it is different).

The problem with Lorentz's time dilation is it was an ad hoc fix to account for the Michealson Morely experiment, unlike Special Relativity which was supposed to be arrived at from "first principles" and thus seemed to be more comprehensive. But I'm yet to see a relativist properly account for the real observed asymmetric behavior. There are excuses but they usually start invoking acceleration, even though the entire equation is with respect to v, clearly.

Regarding the light cone, that seems to be something in General Relativity and I haven't read into that at the same level of detail (or debated it in detail). I was focusing on Special Relativity in my comment. Special Relativity is based on the idea that all motion is relative and there are no preferred frames. So we can only talk about the relative velocity of one frame to another rather than relative to something's own light cone (however they are defining that concept).