I am new to Statistics and was reading about Decision errors in Hypothesis Testing. My question is that why is Type-II error an error at all? From what I understand, it arises when we fail to reject a false null hypothesis. When we fail to reject null hypothesis, it simply means that we do not have strong evidence to reject it. We are not making any comment about which of the two hypothesis is true (or false) . Either can be true. We are not saying that the null hypothesis is true. Then, why is such a conclusion called an error?
Basically, it's called an error because you're making the wrong decision. Like for instance, let's say your null hypothesis is something like: The patient does not have cancer. And the Alternative: The patient does have cancer. A type 2 error in this regard would be like saying "Oh, this patient does not have cancer" while they actually do have cancer. We are FAILING to REJECT a FALSE null. So I guess the most simple way to explain this is, we aren't catching a mistake, and letting it slide by. Of course this isn't all there is to it, but it should help kind of show what a type 2 error is and why it's something you want to avoid.
This type of error happens when you say that the null hypothesis is true when it is actually false. For our null hypothesis that dogs live longer than cats, it would be like saying that dogs do live longer than cats, when in fact, they don't. To help you remember a type II error, think of two wrongs. You are wrongly thinking that the null hypothesis is wrong. The probability of making a type II error is labeled with a beta.
You'll never actually know if you've made a Type-II (or type-I) error in practice. As you surmise, during the hypothesis testing stage you either do or do not reject H0.
Type-I and type-II errors are more useful in terms of error rate. This is a function of your alpha level and statistical power. So for instance you can set alpha to 0.05, 0.01, 0.001, etc based on your tolerance for such errors. But again you can't know after the fact.
So it's called an 'error' from a pedagogical standpoint.