Statistics: Significance Testing: How to think about p-value?

As I embarked on learning more about data analysis with python on a popular online code school, it quickly became apparent that i needed a good primer on statistics (majored in history). And so I've spent a good amount of time here going through the AP stats lessons (which are awesome, thanks!) and think I've come a way with a good conceptual understanding of significance testing (i.e. the probability of getting a given result that would lead to rejecting the null hypothesis therefore suggesting we adopt the alternative hypothesis)

However, the code school prefers to frame significance testing as something like: "We know that the p-value is the probability that we incorrectly reject the null hypothesis on each t-test. "

Is one of these ways of conceptualizing significance tests better than the other or are the complementary?

Thanks.