About an ambiguity that really prevents me from understanding the principle " the laws of physics are invariant in all inertial frames" The principle is often stated as self explanatory .... 
The sentence " the laws of physics are the same in all inertial frames" could mean 2 very different things ( at least from my point of view). 
(1) Does it mean that there is a special class of frames that are "inertial" ( and some that are not) , and that, in this special class, the laws of physics are invariant? 
(2) Or should the property " being inertial"  ( in translation one relatively to the other) " be seen as an equivalence relation that partitions the set of frames, in such a way  that, in each " cell" of this partition, there are invariant laws of physics ( though these invariant laws are not the same for all the classes of frames). 
 A: I am not sure what distinction you are making, but the answer is that your paragraph labelled (1) is correct.
You must bear in mind that reference frames are fictitious abstractions that we overlay on reality to allow us to keep track of distances and elapsed times. Reality remains the same regardless of which frame we superimpose. The question is whether we can use the same mathematical formulae to calculate effects regardless of which frame we overlay. Clearly that is not always true- if you pick two frames, one of which is rotating, moving east and yo-yoing up and down relative to the other, you will need to make some adjustments if you switch from one to the other, as you would if you used a frame with a logarithmic scale, for example. 
A: Take a lift from the first floor to the tenth floor. You feel getting heavier when the lift leaves the first floor and getting lighter when it arrives at the tenth floor. 
You can explain this by noticing that the lift is not an inertial frame. It's accelerated and later slowed down, relative to earth (which is not strictly speaking an inertial frame, but much more so than the lift). 
If you claimed that the lift was an inertial frame, you would have to come up with some really, really strange laws of physics that come into effect at the start and end of any trip in the lift and explain the change in weight.
A: The correct interpretation is (1). The principle states that, when writing the laws of physics using coordinates (spatial and temporal) of  reference  frames, the form of the laws turn out to be  identical in every inertial reference frame.
Invariant is used here as of identical form.
