Does an enumerator print the first occurrence of a word in finite time?
Your interpretation is correct, and the easiest way to look at it is to know both definitions of recursively enumerable sets:


There is an algorithm (potentially running forever) that enumerates the members of $S$.
There is an algorithm for which the algorithm halts only on elements of $S$.


These two definitions are equivalent, and if you use the second one then you can easily answer your own question.

However, do note an important caveat. Although for any given word $w \in S$ there is some time $T_w$ such that after $T_w$ steps, $w$ will have been enumerated. However, there is no way to bound $T_w$ from above as a function of $w$. It can grow faster than any computable function.