I think I understood now. In the graph I posted, some amount of energy was spent (Ea) so that transition state is reached. Then, the same amount of energy is released and an additional amount of energy (that is the energy gap between the reactants and products, or ΔH) is released and the products are made. So, more energy was released than spent, making it an exothermic reaction. All the energy spent is now the activation energy of the reverse reaction.
For endothermic (the graph you posted) Ea was spent, some amount was released, but smaller than the Ea, and the energy gap is ΔH.
This makes both statements true then.