Some (but by no means all) authors define a limit point of a set in such a way as to exclude an isolated point from being a limit point. But when defining a limit point of a sequence, it is essential to allow an isolated point to count as a limit, otherwise a constant sequence would not converge. Lang's definition in comment #3 above refers to limit points of sequences, so of course it allows isolated limit points. I have no idea whether Lang uses the inclusive or the exclusive definition for limit points of sets in metric spaces.
The Wikipedia definition for a limit point of a set is uncompromisingly exclusive: "In mathematics, a limit point (or accumulation point) of a set S in a topological space X is a point x in X that can be "approximated" by points of S other than x itself."