That's what I call a discussion.

Right, I want to summarize now what I see, because I can't really reply to all the minor things that matter in all your comments, but I thank you all anyway.

1. Let's return to the very beginning. What matters here, as I think, is the relativity principle, which in my mind does not reduce to simply the invariance of the physical laws. Still, the foundation of it is that we cannot distinguish between the situation of any inertial observer, i.e. we must be unable to say which one 'really' has moved.

That's why I prefer to go on an assumption (withot insisting it be correct for now) that the spherical flash of light taking place at the origins of the coordinate systems the moment they coincide must be the same for both Observer O and Observer O' as they both perceive it as a flash coming from a motionless source in their respective frames. If one should insist that this is not so, that the uniform flux of rays for Observer O shouldn't be such for Observer O', we could always remind them that the situation is symmetrical, or otherwise there is an observer in absolute motion and an observer at absolute rest. Each observer has had that flash in their frame considering its source motionless, and we might as well start the argument from the point of view of Observer O'. The outcome of this argument must not depend on the starting point (that's another possible formulation of the relativity principle). If we start it from the point of view of Observer O', we'll get the thinning of flux on the opposite side, and I don't know how to proceed from here.

The arguments mentioning aberration imply continuous radiation, but what we deal with here is a singular momentary burst of photons at the same moment of time for both observers, and it will be very difficult to substantiate the divergence of their trajectories in either frame, unless we postulate - not quite clear on what grounds - that for one observer the quality of flux was this, ergo, for the other one it should be that (again, dependence on the starting point).

I'm saying it again I will not insist that the above is absolutely correct. My purpose for now is that the participants see my problem as I see it.

2. I'm not quite convinced about the outcomes of the mirror thought experiment. I understand @Ibix couldn't go into detail for the lack of convenience and paper, but I guess I must insist we talk specifically about the flux of photons, physical objects, and not the characteristics of radiation, since we deal with a limited number of oscillators and at the end of the day it's all that matters. The number of photons flying back past Observer O' must be the same as that flying back past Observer O.

3. For the Minkowski diagram, let me start from afar. Not so long ago I watched a YouTube video where the presenter claimed the invariance of the Einsteinian equations (ct)^2 = x^2 + y^2 + z^2 and (ct')^2 = x'^2 + y'^2 + z'^2 was all wrong, and to prove his point he showed some drawings plotted by maths software where there were ellipses not circles. I knew he was wrong. I saw his mistake immediately, because some time before I had fooled around with a maths soft myself and made it draw ellipses from the Lorentz transforms. This mistake was representing not the view of the moving observer, but instead the view of the moving observer as seen by the stationary observer.

I'm afraid you can't prove anything by slicing the light cone belonging to the one observer by the plane perpendicular to a diffent time axis. The equations (ct)^2 = x^2 + y^2 + z^2 and (ct')^2 = x'^2 + y'^2 + z'^2 are not different slices of the same light cone - they are two different light cones. Indeed, those are equations of 4-cone with ct or ct' for the axis, and the sphere of light is its 3D projection. So slicing the light cone with ct for the axis by the plane orthogonal to ct' is the error of the same nature as the afore-mentioned YouTuber made, representation of the view of the moving observer not as it is but as seen by the stationary observer.

@Ibix wrote: "You can draw the same lightcone and planes in any frame. I want to do it in the frame where the light intensity is isotropic so that I can conclude it is not osotropic in other frames without explicitly doing any maths". - I'm afraid not. You draw a non-synchronized plane, and that testifies to nothing but to the view of O' by O, not the view by O' themselves.

Before we go any further and speak about what aberration in my mind really has to do with all that (a somewhat complicated topic), we need to agree that there is a problem with the light clock.