The laws of physics are the same in all inertial reference frames.
This is a correct principle, but it is useless unless you also state which things are the laws of physics. If they the laws include Newton's laws as usually written then you get Galilean Relativity, if the laws include Maxwell's equation then you get special relativity.
The speed light is traveling at is the same no matter what the constant velocity of the observer (inertial frame reference) looking at the light is.
This is a special case of the first one where the laws of physics include Maxwell's equation, hence the prediction that there are waves travelling g at speed c relative to the inertial frame (Maxwell predicts waves travelling at speed c in the frame in which Maxwell holds).
This is all we need to actually derive the rest of special relativity, so it's a good thing to have. Since it is a special case of the first one and the first one is the correct principle it is redundant to have both.
If I am not mistaken, the observer, whether stationary or traveling at constant velocity, is the inertial reference frame, to which the speed that light is traveling at is relative.
If Maxwell holds in every inertial frame and Maxwell predicts light travelling at speed c in the frame where Maxwell holds then you are wrong. It turns out your opinion about how velocity in one frame looks like to another frame, was just that. An opinion. Baggage in fact.
And you can hold onto the baggage, but if you do so you have to give up the principle of relativity (version 1) or you have to say that Maxwell doesn't hold in evey frame.
Or you can put your cards on the table and see what experiments go along with. If Maxwell only held in one frame then light travels at speed c in just one frame so we should be able to detect that. We try. We fail. Instead, experiments show that for every inertial frame, regardless of velocity, light moves at speed c.
If that is the case then everything else must change, every assumption about how velocity and space and time and relative velocity how hey all work together. Each one must be reexamined to see which ones only held approximately.
If that is correct, the version 1 is contradicting what Einstein said about speed of light being always constant.
Absolutely not. The contradiction comes from your baggage. And if you ignore the fact that you assumed things that are wrong, then contradictions will appear over and over again.
Because the "traditional" law of physics clearly says that there has to be some change in the observed speed of light as the speed of the observer varies.
Newton's Laws don't say that (and Newton's laws do have to be changed). Maxwell's don't say that (and Maxwell doesn't have to be changed). Basically start with Maxwell and trust nothing else and figure out what has to hold if you want Maxwell and version 1. The first thing you get from taking version 1 and taking Maxwell is you get version 2 and from that you can get the rest.
I was saying it would contradict Einstein's own theory if like you said, the "law of physics" were Galilean Relativity. I just wanted to make sure Einstein meant Maxwell when he said the "law of physics".
He called his paper " On the electrodynamics of moving bodies" so he clearly meant Maxwell and clearly wanted the principle of relativity not the specific Galilean frame transformations.
Are both parts of the first postulate of relativity, just that the second version is more specific than the first?
The second version is more what he used, the first is more what he meant and what we think is true. If for instance photons are massive then light goes at different speeds but the speed $c$ is still an invariant upper bound to all speeds that everyone agrees on so relativity wouldn't change. So relativity will be the same even if Maxwell is wrong. The first is correct. And the general transformation allows an invariant speed (for Galilean relativity that speed is infinite) and that speed can be determined by experiment. We've done that and the speed is $c.$ Even if Maxwell is wrong.