Limit Point is defined as:
Wolfram MathWorld: A number $x$ such that for all $\epsilon \gt 0$, there exists a member of the set $y$ different from $x$ such that $|y-x| \lt \epsilon$.
Proof Wiki: Some sources define a point $ x \in S$ to be a limit point of $A$ iff every open neighbourhood $U$ of $x$ satisfies: $ A \cap (U \smallsetminus \{x\}) \neq \emptyset $
What I don't understand is what prevents the above definitions from calling interior points (points which lie in the interior of the boundaries?) as limit points?
For instance, George Simmons defines the sequence $\{1, \frac{1}{2}, \frac{1}{3} \cdots \}$ and states that $0$ is the limit point and $0$ is the ONLY limit point.
If I select $\frac{1}{2}$, every neighbourhood of $\frac{1}{2}$ (minus the point $\frac{1}{2}$) has a non-zero intersection with the set $A$. Why not call $\frac{1}{2}$ the limit point?