How Can I Recognize When to Use ε-Transitions in an NFA?
Your first drawing is a DFA (and thus an NFA) and does match the language described.  The NFA with $\varepsilon$ transitions is a fairly natural (and mechanical) translation of the regular expression, but other than that, it isn't inherently "more correct" than the NFA you started with.

The only confusion I see here other than uncertainty is once you validated that your NFA was a DFA you seemed to have just taken it for granted that it was minimal.  Starting from your DFA, you could have noticed that S4 has the same out-transitions as S2 and that they might possibly be joined.  Doing so would have immediately produced the final DFA.