= Solution
The forcing is <cardinal-preserving> over $M$ when every ground-model cardinal remains a cardinal in every <generic extension>:
$$
\boxed{\forall G\text{ generic over }M\ \forall\kappa\in M\quad
M\models\text{“}\kappa\text{ is a cardinal”}
\Rightarrow M[G]\models\text{“}\kappa\text{ is a cardinal”}.}
$$
<Forcing preserves ordinals>, and an ordinal that was not a cardinal already has a ground-model bijection with a smaller ordinal, which persists. Thus this definition gives equality of the cardinal ordinals in the two models.
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