Why is the p-value defined the way it is (as opposed to a more intuitive measure)? As I understand it the p-value for some hypothesis H and some test-statistic T is defined as the probability of observing a test-statistic T' which is at least as extreme as T, conditional on H being true.
First question: should "at least as extreme" always be interpreted as "the absolute value of T' is at least as large as T"? If not, when shouldn't it be interpreted that way?
Second question: What is the reasoning behind defining the p-value this way as opposed to just the likelihood of the data -- "probability of observing test-statistic T, or a value in the epsilon region around T, conditional on H being true "-- which seems more clearly related to the strength of evidence that T provides with respect to H than the p-value?

How is $p$-value actually defined?
Definition $1.$ (cf. $\rm[I]$) A $p$-value is a test statistic $p:\mathcal X\to [0,1]$ such that $$\mathbb P_\theta (p(\mathbf X) \leq \alpha)\leq\alpha,~~\forall\theta\in\Omega_\mathcal H, ~\forall\alpha\in(0,1).\tag{1.a}\label 1$$
Consider a series of nested tests $\langle \varphi_\alpha\rangle$ in the sense that $\varphi_\alpha(x) \leq\varphi_{\alpha^\prime}(x) $ for $\alpha<\alpha^\prime.$
Define $$\hat p:=\inf\{\alpha:\varphi_\alpha=1\}.\tag{1.b}\label b$$
Observation $1.1.$ $\hat p$ is a valid $p$-value.
Formally, if for a set of nested test functions $$\sup_{\theta\in\Omega_\mathcal H}\mathbb P_\theta(\varphi_\alpha(\mathbf X) \leq \alpha) \leq \alpha ~~\forall\alpha\in(0, 1), \tag 2\label 2$$
then for all $u\in(0,1),$ $$\mathbb P_\theta\left(\hat p\leq u\right) \leq u. \tag 3\label 3$$
$\eqref 3$ is easy to see (cf. $\rm [II]$ ) for $\left\{\hat p\leq u\right\}$ means $\{\varphi_v(\mathbf X) =1\}$ for all $u<v.$ Then, let $v\to u. $
$\blacksquare$
Now consider a test statistic $W(\mathbf X) $ whose large values indicate the rejection of $\mathcal H. $
Observation $1.2.$ (cf.$\rm [I]$) Define
$$p(\mathbf x) := \sup_{\theta\in\Omega_\mathcal H} \mathbb P_\theta(W(\mathbf X) \geq W(\mathbf x)).\tag{1.c}\label c$$ $p(\mathbf x) $ is also a valid $p$-value.
Notice that
\begin{align}p_\theta(\mathbf x) &= \mathbb P_\theta (W(\mathbf X) \geq W(\mathbf x))\\&= \mathrm F_\theta(-W(\mathbf x)),\tag 4\end{align}
which implies $p_\theta(\mathbf x) $ is stochastically greater than or equal to $\mathcal U(0, 1).$ Then as $p(\mathbf x) \geq p_\theta(\mathbf x), $ $\eqref 1$ follows.
$\blacksquare$

When one talks about $p$-value, they are basically meaning $\eqref 1$ or $\eqref c$ which as outlined above are genuine $p$-values.
The phrase as extreme as is essential to define $p$-value. Gloss over the definition and its equivalence as outlined above.
However, what does it imply intuitively? What does lower $p$-value mean? Why is the phrase necessary?
Over the years, there have been many CV posts dealing with the specifics. Please have a look at some of those:
$\bullet$ Why is smaller the p-value, larger is the significance?
$\bullet$ Does p-value ever depend on the alternative? (courtesy Richard Hardy)
$\bullet$ What is the meaning of p values and t values in statistical tests?
and links therein.

References:
$\rm [I]$ Statistical Inference, George Casella, Roger L. Berger, Wadsworth, $2002, $ sec. $8.3, $ pp. $397-398.$
$\rm [II]$ Testing Statistical Hypotheses, E. L. Lehmann, Joseph P. Romano, Springer Science$+$Business Media, $2005, $ sec. $3.3, $ pp. $63-64.$
