Is $E=mc^2$ contradicting conservation of energy?

Question

If we state that, on one hand, energy is conserved because :

$$\Delta PotentialEnergy+\Delta KineticEnergy=0 \tag{1}$$

And we state on the other hand that:

$$Energy=mc^2 \tag{2}$$

Don't we run into a contradiction? As I understand, $E=mc^2$ doesn't work for potential energy (potential energy doesn't show up as mass). Therefore potential energy isn't strictly speaking energy at least in $E=mc^2$'s context. Therefore we can't say that energy is conserved but rather that the sum of kinetic energy and potential energy is conserved.

Answer 1

The formula $E=mc^2$ gives the rest energy of an isolated system. By definition it includes only internal kinetic and potential energy.

Answer 2

In the formula $E=mc^2$, $E$ is the rest energy of the object or system and $m$ is its rest mass. The use of the letter $E$ is misleading because it implies it is the total energy of the object, which is in fact the sum of the rest, kinetic and potential energies. This total energy is conserved, even when energy is transferred between rest energy and other energy stores, e.g. in matter-antimatter annihilation.