If a set is open, does it mean that every point is an interior point?

Definitions should always be treated as "if and only if". So when the text says something like "$E$ is open if every point of $E$ is an interior point of $E$" (I'm guessing it was italicized as such so as to indicate that the sentence is presenting a definition), read:

$E$ is open $\iff E$ every point of $E$ is an interior point of $E$.

Moreover, whenever you have an "if and only if" statement about an object, this statement can be used as a definition for that object. For instance, here are two possible (equivalent) definitions an author could choose for "infinite set" (there are surely many others):

$\bullet \quad$ We call a set $X$ infinite whenever there is an injection $\mathbb{N} \hookrightarrow X$.

$\bullet \quad$ We call a set $X$ infinite whenever there is a nonempty, proper subset $A \subsetneq X$ such that there is a bijection between $X$ and $X \setminus A$.

Such equivalent definitions are one tool that authors can use to motivate and ultimately present the same topic from different perspectives.