What direction of motion?
The one you specified in your thought experiment!
The image you linked to had your cube moving. I assumed you were interested in examining the situation from a reference frame that was watching the cube fly past. Otherwise, what's the point of your thought experiment?
You are IN THE CUBE, and you send a light signal from the center to the receivers. The light sphere contacts the receivers at different times.
In which reference frame?
Your scenario has the light being sent to the 6 faces of the cube from the exact centre of the cube, right? If the receivers are in the centres of all the faces, then they will all detect the light at the same time, in the reference frame of the cube. Why would they not?
In a frame that sees the cube moving, of course the light will reach different faces at different times. Some faces will be moving away from the light emission point while others move towards it.
From those times you can CALCULATE the motion of the cube in space.
If you're outside the cube, yes.
In the cube there is NO MOTION.
The light is in motion. The cube walls don't move.
The motion is only the cube changing position in space, and the coordinates reflect that change in space.
Which coordinates? The ones in which the cube is moving, it seems.
You're very lax about specifying whose coordinates you're using, and about specifying which reference frame you're talking about. If you want to talk about relativity, you need to be specific, because observers in different frames use different coordinate systems.
What you are claiming is that a ball changes shape if you travel real fast past it. That is NONSENSE! The ball does not change shape!
100+ years of experiments prove you wrong, I'm sorry to tell you. Apparently, the universe doesn't care what you consider nonsense. It just does what it does.
That is a circular argument claiming that we have to use length contraction and time dilation to make the numbers add up, because that is the real world is length contraction and time dilation.
It would be circular if we were talking about an imaginary theory that hasn't been tested. But we've thoroughly tested relativity over 100+ years. Relativity is the only theory that can explain the experimental results we see. Newtonian physics is dead. Sorry to break it to you.