Power comparison when alternate hypothesis is true

Question

From what I know, the power of the test is the probability of rejecting $H_0$ when $H_a$ is true. I think I'm getting mixed up with power and $\alpha$, but am I correct in thinking that when $H_a$ is true, we actually want a higher power?

Answer 1

Yes correct.

The power is the probability to Reject $H_0$ when it is false (when is true $H_1$), no error

Type I error, $\alpha$ is the probabilty to reject $H_0$ when it is true (error)

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The probability to Accept $H_0$ when the alternative is true (complement of the power) is "Type II Error"