Einstein's light speed postulate

They are not equivalent:

Just from the fact (or assumption) that there is a maximum signal speed, you cannot conclude that light travels by that speed. It could very well be that light travels at a smaller velocity than $c$, even depending on its frequency, due to some finite photon rest mass.

You can still formulate RT all the same using $c$, it would just be the case that $c$ is not the speed of light but the speed of causality.

I like this, so I'm going to try. I'm going to add in "light transmits signals", since it seems like that equivalency is not what the OP is getting at. Also, I'm going to say "there is a maximum speed for everything". I think those two things come from reasonable definitions of signals, light, and what a "source" is:

1->2: Go to a frame in which the speed of light is maximized. If the speed of light is maximized, it is constant, which implies the speed of the frame must be constant as well (relative to some other frame).

2->1: Say we are in this frame, and the speed of light is c, regardless of the motion of the source. Now consider that the source is some kind of laser, moving in the direction of the beam at a speed v. If v > c, we can't in any reasonable way measure the speed c - it is either equal to v, violating the assumption, or the light doesn't get emitted, implicitly violating the assumption. So (1).

I'm not incredibly happy with that - for example, I would rather have used Newton's first last to get 1->2. But I think it's reasonably good.