= Solution
For parameters $a_1,\ldots,a_n\in M$, <set-theoretic absoluteness> means agreement of the two membership <first-order structures>:
$$
\boxed{(M,\in)\models\varphi(a_1,\ldots,a_n)\ \Longleftrightarrow\ (N,\in)\models\varphi(a_1,\ldots,a_n).}
$$
This must hold for every such parameter tuple. Equivalently, <formula relativization to a class> produces two formulas $\varphi^M$ and $\varphi^N$ whose truth values agree on tuples from $M$. Agreement for parameters only in $N\setminus M$ is not part of the definition. For a proper <class in set theory>, this notation is understood separately for each fixed <first-order formula>, rather than as an unrestricted truth predicate for the entire universe.
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