Let $P$ be a point in 3-space and consider a located vector $ \overrightarrow {0N}$. We define the plane passing through $P$ perpendicular to $ \overrightarrow {0N}$ to be the collection of all points $X$ such that the located vector $ \overrightarrow {PX}$ is perpendicular to $ \overrightarrow {0N}$. This amounts to the condition $(X - P) \cdot N = 0$ or $X \cdot N = P \cdot N$.
My question has to do with $X$. On the one hand, $X$ is a collection of all points for which a certain property is true. So, $X$ is basically a plane. But since $ \overrightarrow {PX}$ is a located vector, $X$ also must be a single point.
Do you think they have used the same variable to denote two different things?
No, $X$ is always just one point in the plane.
The quote does not give any name to the plane (or collection of points) itself.
You should parse it as "the collection of all [points $X$ such that ...]", not "[the collection of all points] $X$".