[whimsicalism]

sounds like you had a chemistry education. relativistic mass is IMO very much not a useful way of thinking about this and it is sort of tautologically true that E = m_relativistic because “relativistic mass” is just taking the concept of energy and renaming it “mass”

[amluto]

This is all sort of silly IMO. The equation, like basically all equations, needs context. What’s E? What’s m? If E is the total energy of the system and m is the mass (inertial or gravitational? how far past 1905 do you want to go?), then there isn’t a correction. If m is rest mass and E is total energy, then I would call it flat-out wrong, not merely approximate. After all, a decent theory really ought to reproduce Newtonian mechanics under some conditions beyond completely at rest.

IMO, when people get excited about E=mc^2, it’s in contexts like noticing that atoms have rest masses that are generally somewhat below the mass of a proton or neutron times the number of protons and neutrons in the atom, and that the mass difference is the binding energy of the nucleus, and you can do nuclear reactions and convert between mass and energy! And then E=mc^2 is apparently exactly true, or at least true to an excellent degree, even though the energies involved are extremely large and Newtonian mechanics can’t even come close to accounting for what’s going on.

[whimsicalism]

inertial mass, rest mass, gravitational mass - these are essentially all the same thing. “relativistic mass” is an additional concept where we rewrite energy as mass and is considered archaic