Rudin defines a limit point of a metric space $M$ to be any point $p$ such that $\forall r > 0,$ there exists a point $y \in M_r(p)$ such that $y \neq p$ and $y \in M.$ Pugh defines a limit point to be "any point that is the limit of a sequence."
Consider the set $\{1,2,3,4\}.$ Under Rudin's definition, these elements are not considered limit points. Under Pugh's, they are (consider the constant sequence). Is it a matter of definition, and does that definition vary with books? Or Have a misunderstood something?
The terminology around relations of points to sets is not completely standardised. Rudin's definition of "$p$ is a limit point of a set $A$" is quite standard; and in terms of general topology is usually formulated as "every neigbourhood of $p$ contains a point of $A$ distinct from $p$". This is denoted symbolically as $p \in A'$ (the set of limit points of $A$ is called the derived set $A'$ of $A$).
If we remove the "distinct from $p$" clause, so that every neighbourhood of $p$ (or metric ball $M_r(p)$ if you prefer) intersects $A$, you get what's usually called an "adherent point of $A$". All those adherent points of $A$ together are denoted $\overline{A}$, and this set is called the closure of $A$, which turns out to be the same as the smallest closed subset of $X$ that contains $A$. It's clear that all points of $A$ itself are adherent points of it, because $p$ will always be in that intersection of a neighbourhood of $p$ with $A$. Also, $A' \subseteq \overline{A}$ by definition, and in fact we can show that $\overline{A} = A \cup A'$ (adherent point are trivial ones (from $A$) plus the limit points).
Also, in a general space we can define the sequential closure of $A$ (no standard notation for that): the set of all $p \in X$ such that there is a sequence $(a_n)_n$ from $A$ such that $a_n \to p$. In general topological spaces, this is no topological closure operation (because the sequential closure of the sequential closure of $A$ can be larger than the sequential closure of $A$ itself). In metric spaces (or more generally first countable ones) it turns out that the sequential closure of a set actually always equals the closure of that set, so that every adherent point of $A$ is a limit of a sequence from $A$ (possibly a constant one, in case of $p \in A$ in your example).
So Rudin defines a limit point classically, while Pugh defines it as "an adherent point via sequences" and this only really "works" in an analysis context, where we work in metric spaces. In general it's good to be aware of differences in terminology between different authors and texts. Always check the definitions of things like "limit point", or "accumulation points". These terms also can have a different meaning when defined for sequences instead of sets, e.g.