Generalized Büchi Automata - Formal definition of a state appearing infinitely often?

Question

I am studying generalized Büchi automata and I don't really understand when a state is considered to appear infinitely often. The definition I have is:

A state $s$ appears infinitely often if there exists an infinite set of points $i \in N$ such that the $i$th state of the execution is $s$.

But there's also an example which I think contradicts this definition.

According to the example, the language accepted by this automaton is the language where the string $ab$ appears infinitely often.

Why isn't it just $a$ appearing infinitely often? State 2 would be reached even if we only had $a$ as input. Which is wrong, the example or the definition? Or did I misunderstand the definition?