Does "feature importance" depend on the model type?
When it comes to feature importance I always go with a model-agnostic measure, as you well mention if you have two different models, they will interpret importance in different terms (Linear models as the coefficient and Tree-based models as the information gain/impurity decrease on each feature.
So you already mention one measure that does not depends on the model, rather on the metric you are interested in; Permutation importance does not care about what model you are using, but the impact that a feature has on the global performance.
This reference might give you a better idea of the advantages of using permutation importance over tree-based models importance Permutation Importance vs Random Forest Feature Importance