Error in Hypothesis Testing

There are two types of errors that can occur when we make a decision in statistical hypothesis testing. The first is a Type I error, which means we reject H0 when it is really true. Another error is called Type II error, when means we accept H0 but it is really false.

I am always confused about these two, despite many years of statistical background :(

In short,
Type I Error:  Incorrectly reject the H0
Type II Error: Incorrectly NOT reject the H0

We denote the symbols for the probability of Type I error is $\alpha$ and of Type II error is $\beta$.

The $\alpha$ is sometime called the sinificance level of the test and it represents the maximum probability of Type I error that will be tolerated. Typical values of $\alpha$ are 0.01, 0.05, 0.10.

I am getting more understanding about Type I and Type II error and it can be linked to the True Positive, False Positive, False Negative and True Negative as follow.

Actually, the main confusion came because of the name "positive". Here, "Positive" means "rejecting H0"  and "Negative" means "fail to reject H0" OK?

Therefore, The 2x2 matrix of test/actual conditions is

Moreover, while Type I error is the False Positive (FP) but the $\alpha$  is not the same as "false positive" as most references try to say. It should be "false positive rate" FP/(FP+TN). This way, it makes more sense for the $1-\alpha$ to be called the specificity = TN/(FP+TN).

On the other hand, Type II error is the False Negative (FN) but the $\beta$ is the "false negative rate"  FN/(TP+FN). Now, the power or $1-\beta$, which is also called the sensitivity, is TP/(TP+FN).