Less wrong, but still off. The p-value is the probability of committing a type-1 error (false positive: we declare X to be president when he really isn't) given a sample and a null hypothesis. P-values are not the 'chances' of seeing the given data, it is a test-statistic that describe how reliable your probability estimates are for a given sample and null.
Actually I think we're still a little off.
The p-value is the probability of seeing a test statistic at least as extreme as the observed sample statistic, under the assumption that the null hypothesis is true. This can be restated in terms of tests and errors.
The p-value is the probability of committing a type-1 error for a statistical test, where the threshold of that test is chosen to be equal to the sample statistic. In more intuitive terms, it's the probability of a type-1 error under the null hypothesis for the most extreme test that the sample data is able to pass.
In your explanation, where you said "given a sample" you should say "given a sample size", to make it clear that the type-1 error probability is not conditional on the sample data. If you condition on the observed data, then the probability of passing a statistical test is going to be 1 or 0 depending on whether or not the data passes the test. It is the test itself, not the error probability, that depends on the sample data. Which is also something you should have specified.
This is the clearest answer I've read so far. The p-value is the minimum probability of ending up with a test statistic that lies in the rejection region (for a given null hypothesis and critical value).
Please re-read onetwofiveten's explanation. All I've done is paraphrase it. Alpha is just an arbitrary cutoff p-value corresponding to a threshold t-value for a given null hypothesis.
You've got it backwards. The p-value is the chance that we see the given data given that X is not president.