I understand that the p-value is defined as the probability to obtain a "more extreme" value of $w$ if $H_0$ is true, i.e.
$p=P(|W| > |w| \ |H_0 $is true$)$
and the "significance level" $\alpha$ is a threshold to decide if it is "admissible" that $w$ comes in null hypothesis setting, i.e.
$p < \alpha \Rightarrow$ reject $H_0$.
However, It is hard for me to understand the meaning of hypothesis test when $\alpha$ is interpreted as a probability, specifically:
$\alpha=P($reject $H_0 | H_0 $is true$)$.
In this case, I will have:
$P(|W| > |w| \ |H_0 $ is true$) \ < P($reject $H_0 | H_0 $is true$)\Rightarrow$ reject $H_0$
Therefore I'm saying that, if $H_0$ is true, the probability of $w$ is less than the probability of rejecting $H_0$, therefore reject $H_0$. Since, for example, I obtained $w$, its probability is lower than the probability of in rejecting $H_0$, so reject. In other words, it is more probable that I reject when I shouldn't, respect to the fact that $w$ has been sampled, so reject. This seems to me that it is more probable that I'm making a reject when I shouldn't, or there is something that escapes me. Furthermore, I'm comparing two different distributions, i.e. $P(W)$ and $P(\text{ reject } H_0)$, that seems to me two different objects, therefore I don't understand the point of comparing them.
This question can be viewed similar to The rationale behind the "fail to reject the null" jargon in hypothesis testing?, but in my opinion is different (it regards also the type II error, which is not the subject of my question) and also to What is the meaning of p values and t values in statistical tests? (which see $\alpha$ simply as a threshold, and not as a probability): Maybe a similiar question is intuition/logic behind comparing p-value and significance level but the given answer is not fully satisfactory for me, and the question has been closed since considered too similar to the other questions.
So, what is the logical meaning in rejecting the null-hypothesis when $\alpha$ is viewed as the probability to make a Type I mistake?