A salt is a compound composed of two charged groups. One positively charged, one negatively charged. For drugs such as adderall, one is a positively charged amphetamine (with an extra hydrogen as a proton) and the other half is a negatively charged counter ion (either sulfate, sacharate or aspartate). The reason it is given as a salt, is that charged molecules dissolve more easily in water. If it was not charged, it would not dissolve and be less effective when taken orally.

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In the case of lithium, it is usually lithium cations (positively charged) with carbonate, bromide or other counter anions. The truth is, we don't really know exactly how lithium works in the brain. It interacts with a number of things but we can't pin down which affects particularly help with mood stabilization.

We colloquially refer to sodium chloride as “salt” and it leads many people to associate the word salt with food. For everyday purposes, that meaning is perfectly fine. However the word “salt” as used in chemistry simply means a neutrally charged compound consisting of positive cation(s) and negative anion(s). Sodium chloride (NaCl, or Na ^+ Cl ^- ) meets this criteria, hence it being called salt.

The same chemical definition fits amphetamine salts and the formulation of lithium used clinically, however that’s the end of their similarity with food items. When using these medications, it isn’t the fact that patients get a “salt” that’s causing a clinical change, but rather that the medications have specific mechanisms of action (e.g. Lithium reduces excitatory stimulation of dopamine and glutamate and up-regulates GABA, though it’s actual mechanism of action is largely unknown).

That doesn’t mean that diet could not, or does not, affect mental health disorders, just that diet’s effect is a completely different topic.