Is the p-value the probability that the null hypothesis is true?

No. A p-value is conditional on the null hypothesis being true. It asks how unusual the observed test statistic would be under that assumption.

A better exam-safe phrasing is:

The p-value is the probability of observing a test statistic at least as extreme as the calculated statistic, assuming H0 is true.

So a p-value of 0.03 does not mean there is a 3 percent probability that H0 is true. It means the observed evidence would be unusual under H0. If alpha is 5 percent, the analyst rejects H0 because 0.03 <= 0.05.