Two conflicting definitions of "$\omega$-accumulation point"

Question

I came across two different definitions of $\omega$-accumulation point.

1. From Steven A. Gaal's Point Set Topology, p.136 [emphasis added]

   Call a point $x \in X$ an $\omega$-accumulation point of $S$ if $N_x \cap S$ is uncountable for every neighbourhood $N_x$.

2. From John L. Kelley's General Topology, pg.137 [some emphasis added]

   A point $x$ is an $\omega$-accumulation point of a set $A$ iff each neighborhood of $x$ contains infinitely many points of $A$.

The second definition is precisely the known definition of an accumulation point. Then what's the point of adding the letter $\omega$ to it?

And these definitions are very different from each other. When the definition is not given which should one assume? And why are there two completely different definitions of the same term?