This is perhaps as much a question of semantics as of physics but it is something I have been thinking about recently and was wondering if anyone else had a perspective on this. Now, it could be that it turns out that this is something that is obvious in theory and I am just being stupid.
We are told that the invariance of the speed of light in vacuum is an axiom, the way there are Axioms in Euclid's geometry. It is a given and there is no point in asking the question, why is the speed of light invariant and constant across frames of reference.
My point is that the invariance of the speed of light and the constancy of the speed of light are different kind of things. And while it is philosophically acceptable to just "know" that the speed of light is constant but it not to just "know" that it is invariant. Fixed constant values such as the mass of an electron or the spin set of an electron are things one can accept as given. That's just the starting condition. Similarly it is understandable that the speed of light is some constant value or as you do in field equations, just say that $c=1$.
However the invariance of the speed of light across reference frames is different. It seems like an obtuse boundary hiding some physics that we cannot yet understand. (Perhaps there are some theories in particle physics I am not aware of).
So my question is this.... is this a semantic game, is saying that the speed of light is fixed the same as saying that it is invariant across reference frames? Because perhaps, something can be constant but appear to be variant (changing) across reference frames because of some co-ordinate transformation. The way gravity does. The way we differentiate real gravity from accelerated reference frames by potentially trying to look for gravitational waves (theoretically). I know this may sound silly but I hope you see what I am trying to say.