Is this a good definition for $A\neq B$ ?
I define $A=B$ to be
$\forall x[(x\in A)\leftrightarrow (x\in B)]\label{1}\tag{1}$
If ($\ref{1}$) is my definition for "$=$", then its negation is ($\ref{2}$)
$\exists x[(x\notin A)\oplus(x\notin B)]\label{2}\tag{2}$
My question: Is ($\ref{2}$) a good definition for $A\neq B$ ? If it is not, which out of ($\ref{1}$) and ($\ref{2}$) is wrong?
($\ref{2}$) seems to say that the elements of A are not guaranteed to be the same as B. It does not appear to say that they will never be the same no matter which element we pick. This is what confused me about the definition, for I have sometimes thought of $\neq$ as meaning that they are never the same.
As a bonus question; would the negation of ($\ref{3}$) have any meaning?
$(A=B)\leftrightarrow(\forall x[(x\in A)\leftrightarrow (x\in B)])\label{3}\tag{3}$
The sane thing is to define $A\ne B$ to mean $\neg(A=B)$.
If you unfold the definition of $A=B$, this becomes $$ A\ne B \quad\text{ means }\quad \neg \forall x(x\in A\leftrightarrow x\in B) $$ There are various things this is logically equivalent to, but that's not a matter for the definitions themselves -- it's just something you might need to appeal to when you reason with those definitions.
Your (2) just says that there is at least one thing that is in one set and is not in the other. That's a perfectly good characterization of disequality between sets.
For example, $\{1,2,3\}$ and $\{2,3,4\}$ are different sets, even through $2$ and $3$ are in both of them, and $6$ or $\pi$ is in neither.