a p-value can be interpreted as the probability of seeing the same response given that the null hypothesis is true.
Not quite: the p-value is the probability of observing a test statistic that is as or more extreme than the one you actually observed. This is why the p-value is expressed as a probability in terms of an inequality: $p = P\left(\Theta \ge \theta\right)$, $p = P\left(\Theta \le \theta\right)$, or $p = P\left(|\Theta| \ge |\theta|\right)$.
However, the significance level, alpha, we know corresponds to the probability type 1 error.
Almost: $\alpha$ is a researcher choice, and represents the willingness of the researcher to falsely reject a single null hypothesis, assuming that the null hypothesis is true.
In the frequentist world it makes sense to reject test statistics that are too unlikely... too extreme assuming the null hypothesis is true. The reasoning is along the lines of "if this test statistic, $\theta$, was so unlikely to be observed if the null hypothesis is true, then perhaps the null hypothesis isn't true!" The p-value is the measure of "how extreme" under the null.
The value of $\alpha$ indicates the researcher's line in the sand: probabilities of observing test statistics as or more extreme as $\alpha$ (because they are smaller than $\alpha$) are rejected because both probabilities are defined under the assumption that the null hypothesis—including the distribution implied by it—is true.