While you are right that you need to be encoding the right details, I disagree on the tests enforcing a design point.

As part of the proper testing strategy, you will have tests that cover individual behavior of a small block/function (real "unit" tests), tests that cover integration points only up to the integration itself, and a small number of end-to-end or multi-component integration tests.

Only the last category should stay mostly idempotent under refactoring, depending on the type of refactor you are doing.

Integration tests will obviously be affected when you are refactoring the interfaces between components, and unit tests will be affected when you are refactoring the components themselves. Yes, you should apply the strategy that keeps it under incremental reverse TDD approach (do the refactor and keep the old interface, potentially by calling into new API from the old; then in second step replace use of old API as well, including in tests).

Tests generally define behavior and implementation in a TDD approach: it'd be weird if they do not need changing at all when you are changing the implementation.