The phrase relativistic energy generally means "energy, taking relativity into account", or it means the contribution of relativity to some type of energy, e.g., to kinetic energy. The formulas for determining these are clear, but a terminology is not unanimously used.
The term mass (i.e., with no qualifying adjective) has come to be used two ways in light of relativity: to mean the object's rest mass, which is the mass as we normally think of it, or to mean a relativistic mass, the mass equivalent of the object's total energy, including kinetic energy and relativistic effects. (Effectively, this potential terminology-confusion only arises due to relativity, and the difference is only for a moving object, and significant only if at relativistic speeds.)
Some formulas for energy that take relativity into account, specifying which mass we mean here:
rest energy = MrestC² total energy aka total relativistic energy = γMrestC² kinetic energy = (γ-1)MrestC² relativistic contribution to kinetic energy = ((γ-1)C²-V²/2)Mrest Mrel = γMrest rest energy = MrelC²/γ total energy = MrelC² kinetic energy = (1-1/γ)MrelC² relativistic contribution to kinetic energy = ((γ-1)C²-V²/2)Mrel/γ