Would it be safe to say that mass is kind of a property of energy? In a video i just saw about the true meaning of $E=mc^2$, it said something that really got me thinking; "Mass is not really converted to energy" and that mass wasn't actually a thing, but more of a property that energy exhibits, so that mass is not so much of an indicator of stuff (matter), but more of an indicator of amount of energy manifested in different ways, which are kinetic and potencial energy. 
 A: Let me start by examining your last line about kinetic and potential energies. Potential energy is NOT a property of matter. But kinetic energy IS! 
Here is a short derivation of why kinetic energy is a property of matter: $E=mc^2$ in rest frame of the particle. But if you switch to a different inertial frame in which the particle is moving at speed v, $E' = \gamma mc^2$ where $$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$ 
If you take the limit where $v<<c$ (which is the case for everyday physics), and do a taylor expansion on $\gamma$, you obtain $\gamma \approx 1+\frac{1}{2}\frac{v^2}{c^2} $
Then $E' = mc^2 + \frac{1}{2}mv^2 $!! This is how the kinetic energy arises from Einstein's mass-energy relation. Potential energy cannot be derived from mass-energy relation since potential energy involves INTERACTIONS between particles instead of particles themselves. 
Going back to your question, I think that instead of saying "mass is a property of energy", it is more appropriate to say that "mass and energy are both properties of particles". The specific classifications of energy are contrived and designed to indicate "manifestations" of energy in different forms. But fundamentally, mass and energy are equivalent. 
