Answer should be in short and simple ....even a layman should understand it. Don't cut and paste, like research papers published in very high quality and top journals. Write your own understanding about the phenomenon here.
Preston Guynn. Thanks for pointing out my past activities.
From now onwards, let us think independently based on our scientific knowledge acquired from past and present learning and observing the present natural phenomena. Like Galileo Newton Einstein..., let many of us become scientists to meet today's challenges and develop our world society.
The mass of a relativistically moving rigid body is Lorentz invariant, that is, unchanged. It does not depend on the speed of movement of the body, so it does not change when moving from one frame of reference to another.
Regarding the equations you wrote down:
[I} equation m = γ* m0 :
A). is incorrect, because if m in this equation means relativistic mass, then such mass does not exist in nature.
B). The notation γ* m0 itself is correct, but it does not represent relativistic mass, but represents the total relativistic inertia I of a moving body.
C). Based on A) and B) the only correct entry is I = γ * m0 ;
[II}. equation E = m * c ^ 2
A). You have correctly identified that this equation is incorrect. It is always false, because in it m denotes a non-existent relativistic mass.
B). This equation is also incorrect because no one has ever been able to prove it. Neither A. Einstein (there are very weighty arguments on this matter), nor anyone else.
WITH). The correct equation is Eо = m * c ^ 2 (here o is the subscript) for energy and rest mass (but not for motion!). Few people know that A. Einstein himself was unable to logically substantiate this equation (there are very weighty arguments in this regard). The validity of this equation was proven later (after Einstein) by M. Planck in 1907 and P. Langevin in 1913,
D). Based on A) - C), instead of the equation E = m * c ^ 2, the correct equation is E = Iα * c ^ 2 (where α is the subscript, I denotes the total relativistic inertia of the moving body.
On question Iα, everyone can read my article “On the theory of inertia and the law of equivalence of relativistic inertia and energy” here on ResearchGate, or read my monograph “Is the speed limit of light due to the growth of relativistic inertia? Isn’t Nature laughing at us?”