A basic question in the definition of limit point

A basic question in the definition of limit point

For any subset of $R$ with the usual distance metric, any point inside it
is a limit point. Only when the set is discrete there may be a point
inside it which is not a limit point. Is this correct ?
Another question, according to this definition in the $(0, 1)$ open
interval any point inside it is a limit point. What is the meaning of, for
example, $\frac{1}{2}$ is a limit point of $(0,1)$. I understand the
meaning that $0$ and $1$ is a limit point of $(0,1)$ in the sense that
each of them approximates the one side of $(0,1)$.
BTW, I am new to analysis