Are these statements about p-values correct?

I've seen the following equivalent statements about p-values:

• "It is the probability of wrongly rejecting the null hypothesis if it is in fact true." 

• "The P value or calculated probability is the estimated probability of rejecting the null hypothesis (H0) of a study question when that hypothesis is true." 

Some people I know insist these are wrong, and say these are equivalent to type I error. But the type I error is in fact determined by the significance level. Are these right or wrong?

• This previous discussion might help. – Julian Schuessler Mar 6 '14 at 20:42

There are two p-values of interest. The critical p-value, also known as the $\alpha$-level or significance level, is decided and fixed before the study / analysis is performed. This critical p-value is in fact the probability of Type I error. Your examples are talking about the critical p-value, and I agree with those who are saying they are incorrect or at least misleading, because in common usage "the p-value" refers to the next type of p-value.
We reject the null hypothesis if the observed p-value is less than or equal to the critical p-value. This is because, under the null hypothesis, the observed p-value is uniformly distributed on $[0,1]$. Thus the probability that we reject the null when in fact the null is true is equal to the critical p-value.