No you can't. There are no absolute velocities. All velocities are a relation to something else and light can't be that something else.
But you can use light to determine your absolute angular velocity! That's why the Laser Ring Gyro works. And that's why Foucault and Michelson were able to use the spinning mirror method.
What result do you get with your laser ring gyro on the equator? It tells you nothing, the point you seem to miss is that the laser ring gyro is a third body that you are referencing from, just like my example of the two men floating in space with a spaceship. If you assume that the spaceship is stationary, then you can determine the velocities and directions of the men with respect to the spaceship. The laser ring gyro is no more fixed than the spaceship.
So can you at least give me one good reason why you won't answer any of my color-coded questions?
I have answered those questions. You can answer those questions yourself if you can figure out what your reference frame is and what you mean by an 'instant' in time? When the light leaves the object or when the light is seen by the observer. You don't seem to understand that when you are looking at something you are seeing into the past.
There is no universal 'now'. Time and distance are variables. You should read up on the Michelson-Morley experiment, they tried to use light to determine the Earths velocity and failed. If you were correct that light can determine absolute angular velocity, Michelson and Morley would have succeeded : )