If energy density relates to rest mass density as $\epsilon = \rho / c^2$. How internal energy relates to energy density? Also, when do we write $\epsilon = \rho / c^2$ an when $\epsilon = \rho / c ^2 + €$?
The total energy density is the sum of all the individual types of energy density, for example: rest-mass, (or thermal/`pressure'), internal energy, but may also include electromagnetic, kinetic, etc
i.e. (Eq. 1) $$\epsilon = \rho c^2 + € + ...$$ Note that it is mass-density times $c^2$, not divided by.
If some of the terms on the right-hand-side are zero (like velocity, magnetic field, or internal energy) then we can simplify to,
(Eq. 2) $$\epsilon = \rho c^2.$$ Equation (2) is a special case of equation (1).