None of this either/or reasoning is correct, in my experience. In practice, I write tests both before and after implementation, for different reasons. In practice, my tests both test correctness, and of course they also work as regression tests.

Writing before the fact allows you to test your mental model of the interface, unspoiled by having the implementation fresh in your mind. (Not entirely, since you probably have some implementation ideas very early on.)

Writing tests after the fact is what you must do to explore 1) weak points that occur to you as you implement, and 2) bugs. After-the-fact testing also allows you to hone in on vagueness in the spec, which may show up as (1) or (2).