What does it mean for a nucleus to be more stable? When we say that a nucleus is more stable than another nucleus, are we saying that the nucleus is at a lower energy level than the other nucleus? In chemistry, a more stable compound is one that is at a lower energy level, is this the same when we talk about a nucleus?
If yes, wouldn't this make all nuclei with a larger mass more unstable since they will be at a higher rest mass energy.
But clearly, this is not the case as the binding energy per nucleon graph increases initially before decreasing, suggesting that a heavier nuclei can be more stable than a lighter one.
 A: Wnen a nucleus (or a particle) is called stable, it usually means that it does not decay, i.e., it is not radioactive. In practice one would describe as stable the nuclei and particles that decay over very long times, so long that their decay rate can be considered negligeable for practical purposes. The stabilities of two nuclei are compared in the same sense: the more stable nucleaus is the one having longer lifetime (smaller decay rate).
This is not quite identical to the stability of chemical compounds. The nuclei can be often treated as isolated, so that the value of their decay rate is dominated by the strength of their coupling to the environment. Temperature and hence the excitation energy play minor role in this respect (although they certainly matter, e.g., when considering many nuclei in a nuclear reactor). On the other hand, chemical compounds are nearly always a part of a solution or a gaseous mixture, and their decay rate varies significantly with temperature according to the activation law:
$$
\Gamma\propto e^{-\frac{E_b}{k_BT}},
$$
where $E_b$ is the binding energy.
A: Your reasoning based on the ‘rest mass energy’ is not really relevant to the analysis. On earth, a nucleus at higher altitude would hold more gravitational potential energy. But I don’t think you would intuitively imagine that this would make it less stable. My point is, you could define many different kind of energies/potentials in which you put your nucleus, that doesn’t make it relevant to its stability.
Instead, just recall a very basic idea of the physics of what you are studying to guide your choice of a relevant quantity  . The protons and neutrons are held together by the nuclear force. So what you want to minimize is the potential due to that force. Gravitational fields would be irrelevant to describe how strongly the nuclear force holds the nucleus together, right? Similarly, the mass energy doesn’t really help to describe it.
So the energy that needs to be smallest is the one due to all interactions between nucleons. If this energy of interaction is smaller when the nucleons are packed together vs when they are apart, then they wanna stick together. The more negative that difference is, the more stable the nucleus (i.e. the more energy needed to take it apart)
