Both explanations are valid, which doesn't mean they are contradictory.
It's kind of like one person in the train sees the person on the ground moving, even though from the person on the ground's point of view, it's the person on the train who is moving. Who's moving? Well, the interpretation depends on the referential you choose.
So here, you can say there is a referential "mass" and a referential "energy". They will give seemingly different interpretations, but which are actually equivalent.
(On a side note, there are indeed no differences between E=mc^2 and m=E/c^2, it's just that it's easier to conceptualize inertial mass as being energy at rest with the latter.)