[nils-m-holm]

Which sounds like a fancy way of saying that "there is a 5% chance of the data being noise (while the null hypothesis is true)".

[imh]

"there is a 5% chance of the data being noise" is a statement on the probability of whether the phenomenon is not real given the data: p(it's just noise | data). A p value is a statement on the probability we'd see this data given that the phenomenon isn't real: p(data | it's just noise).

Now, the exact values of a given data set are vanishingly unlikely, so we actually ask about the probability we'd say this data or even more extreme data given that we're just looking at noise. That's a typical p value.

[nils-m-holm]

That's what I meant: P(data | H0). So there is a 5% probability for the data to arise by chance (being noise) given H0 is true.

The whole terminology with values being "extreme" is just unintuitive and unwieldy, IMO.