Why are non-conservative tests desired over conservative ones? In my textbook, it says that P(Type I Error) =  α is a non-conservative test and my professor noted that this one is desired over the conservative test with P(Type I Error)  ≤ α. Why is that the case? Wouldn't we want the probability that we reject $H_0$ when it is true to be less than the significance level?
Edit: Conservative tests will detect a correct violation of the null hypothesis less often than a non-conservative tests (taken from the book).

I suppose your text is actually defining: $P(\text{Type 1 Error}) < \alpha (\text{nominal})$ as conservative and $P(\text{Type 1 Error}) \ge \alpha (\text{nominal})$ as non-conservative. So your definition in sentence two is not aligned with your edit: you should make the inequality in sentence two strict.
You might rather define $P(\text{Type 1 Error}) = \alpha (\text{nominal})$ as a test that is well conserved or of the correct size or achieves nominal alpha level or is correct. All these terms have been used from time to time.
There is no value in having a test that is anti-conservative.
