In Rudin, a limit point $p$ of a set $E$ is defined (paraphrasing) in this way: for any $\epsilon > 0$, $N_{\epsilon}^* (p) \cap E \neq \emptyset$, where $N_{\epsilon}^* p$ is the deleted neighborhood of radius $\epsilon$ of $p$.
In Tao, a limit point of a sequence is defined in this way: $p$ is a limit point of a sequence $(a_n)$ if for any $\epsilon > 0$ and $N \in \mathbb{N}$, there exists $n \geq N$ such that $|a_n - p| \leq \epsilon$.
I assume these definitions are equivalent and it doesn't make much difference that one is defined in terms of a set and the other in terms of a sequence. I am not able to understand intuitively why this is the case, though. Which of these is the "standard" definition? I've seen a number of different terms -- isolated point, accumulation point, contact point, and limit point -- that seem to have conflicting definitions and to sometimes depend on the author.
Take a sequence and regard it as a set. If you already know that for all epsilon and N there exists an $n>N$ such that $|a_n -p| < \epsilon$ and p is not in the sequence itself, then the punctured neighborhood of p of radius $\epsilon$ intersects $E$ in at least that $a_n$. So $p$ is also a limit point in Rudin's definition.
For the other direction, $E$ might be uncountable so it is a more general definition.
Another difference: suppose $a_n=p$ for all n. Then in Tao's definition, p would be a limit point. But in Rudin's definition: because $p$ was deleted, p would not be a limit point.