In other words, "z sigma" means: That a result like this occurs as a statistical fluke, is just as likely as a standard-normal distributed variable giving a value above z.
I would add: If the null hypothesis is true, then "the result like this... (in this case the null hypothesis is of cause that the standard model is true)
If the null hypothesis were true, and the experiment were repeated infinite number of times with a different sample each time then "the result like this or more extreme ...
I agree with adding the "more extreme" part, but I'm not so sure about the infinite number of times part. Certainly, the p-value is (roughly speaking) the probability of seeing a result at least as extreme as the observed result, under the null hypothesis. But one doesn't really need to introduce hypothetical infinite sequences of replications to make sense of that definition.
