I am bit diffident to revive the old topic, but I jumped from herehere, so I post this as a response to the question in the link.
The p-value is a concrete term, there should be no room for misunderstanding. But, it is somehow mystical that colloquial translations of the definition of p-value leads to many different misinterpretations. I think the root of the problem is in the use of the phrases "at least as adverse to null hypothesis" or "at least as extreme as the one in your sample data" etc.
For instance, Wikipedia says
...the p-value is the probability of obtaining the observed sample results (or a more extreme result) when the null hypothesis is actually true.
Meaning of $p$-value is blurred when people first stumble upon "(or a more extreme result)" and start thinking "more extreeeme?".
I think it is better to leave the "more extreme result" to something like indirect speech act. So, my take is
The p-value is the probability of seeing what you see in a "imaginary world" where the null hypothesis is true.
To make the idea concrete, suppose you have sample x consisting of 10 observations and you hypothesize that the population mean is $\mu_0=20$. So, in your hypothesized world, population distribution is $N(20,1)$.
x
#[1] 20.82600 19.30229 18.74753 18.99071 20.14312 16.76647
#[7] 18.94962 17.99331 19.22598 18.68633
You compute t-stat as $t_0=\sqrt{n}\frac{\bar{X}-\mu_0}{s}$, and find out that
sqrt(10) * (mean(x) - 20) / sd(x)
#-2.974405
So, what is the probability of observing $|t_0|$ as large as 2.97 ( "more extreme" comes here) in the imaginary world? In the imaginary world $t_0\sim t(9)$, thus, the p-value must be $$p-value=Pr(|t_0|\geq 2.97)= 0.01559054$$
2*(1 - pt(2.974405, 9))
#[1] 0.01559054
Since p-value is small, it is very unlikely that the sample x would have been drawn in the hypothesized world. Therefore, we conclude that it is very unlikely that the hypothesized world was in fact the actual world.
