Examples of almost compact spaces.

Let $Y=\beta \Bbb N \setminus \{p\}$ for $p \in \Bbb N^\ast$. Then theorem 6.4/6.7 from Gillman and Jerrison tells us that $\beta Y=\beta \Bbb N$ and another standard theorem tells us that the one-point compactification of $Y$ also equals $\beta \Bbb N$. So $Y$ is almost compact.

8L in that book also shows that $\Omega$ (the square of $\omega_1 +1$ with $(\omega_1, \omega_1)$ removed), is another example of an almost compact space, as is the Tychonoff plank, which is closely related. They are introduced in excercise 10R as spaces with a unique uniformity and the non-compact ones are characterised as those Tychonoff $X$ that have $|\beta X\setminus X| =1$.

Concerning the third bullet: It should definitely read as "whenever X is extremally disconnected and compact". Otherwise, $\mathbb{N}$ is extremally disconnected, but $\mathbb{N} \setminus \{1\}$ is not almost compact. Furthermore, if $p$ is isolated in $X$, then $X \setminus \{p\}$ is compact, hence not almost compact as defined above. (Although, almost compactness should include compactness, which actually is, to my knowledge, the most common understanding.)

Anyway, if $X$ is extremally disconnected, compact and $p\in X$, $p$ not isolated in $X$, then $X \setminus \{p\}$ is almost compact:

We have to show, that $\beta (X \setminus \{p\}) = X$, i.e. that if $A, B \subset X \setminus \{p\}$ are completely separated in $X \setminus \{p\}$, then $A, B$ are completely separated in $X$:

There exist open, disjoint subsets $U, V$ of $X \setminus \{p\}$ such that $A \subset U$ and $B \subset V$. $U, V$ are also open in $X$. Since $X$ is extremally disconnected, the closures of $U, V$ are disjoint and compact, hence completely separated. Hence also $A, B$ are completely separated in $X$.