Not rejecting H0(Null Hypothesis) when H0 is False is Type 2 error?

Question

Why failing to reject a null hypothesis $H_0$ is type 2 false negative?

In my understanding:

1. False Positive (Type 1) is judging a false statement incorrectly as true.
2. False Negative (Type 2) is judging a true statement incorrectly as false.

Suppose a medicine Y is effective but X is placebo that has no effect. The null hypothesis $H_0$ will be (Y is X) meaning Y is as ineffective as placebo, and $H_0$ is a false statement.

Then, judging $H_0$ incorrectly as true should be type 1 false positive. However, it is false negative.

• Type II error

The second kind of error is the mistaken failure to reject the null hypothesis as the result of a test procedure. This sort of error is called a type II error (false negative)

Why is judging the alternative hypothesis $H_a$, which is a true statement (Y is NOT X), as false not type 2 false negative?

Please help understand why.